diff --git "a/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt" "b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/ENE0T4oBgHgl3EQfywJf/content/tmp_files/load_file.txt" @@ -0,0 +1,641 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf,len=640 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='02663v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='GR] 6 Jan 2023 ON THE CHARACTERIZATION OF ALTERNATING GROUPS BY CODEGREES MALLORY DOLORFINO, LUKE MARTIN, ZACHARY SLONIM, YUXUAN SUN, AND YONG YANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Define the codegree of χ ∈ Irr(G) as cod(χ) := |G:ker(χ)| χ(1) and denote by cod(G) := {cod(χ) | χ ∈ Irr(G)} the codegree set of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let An be an alternating group of degree n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this paper, we show that An is determined up to isomorphism by cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Introduction Let G be a finite group and Irr(G) the set of all irreducible complex characters of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' For any χ ∈ Irr(G), define the codegree of χ as cod(χ) := |G:ker(χ)| χ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then define the codegree set of G as cod(G) := {cod(χ) | χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The concept of codegrees was originally considered in [8], where the codegree was defined as cod(χ) := |G| χ(1), and it was later modified to its current definition by [22] so that cod(χ) is the same for G and G/N when N ≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Several properties of codegrees have been studied, such as the relationship between the codegrees and element orders, codegrees of p-groups, and groups with few codegrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The codegree set of a group is closely related to the character degree set of a group, which is defined as cd(G) := {χ(1) | χ ∈ Irr(G)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The relationship between the character degree set and a group’s structure is an active area of research – many properties of a group’s structure are largely determined by its character degree set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In 1990, Bertram Huppert made the following conjecture about the relationship between a simple group H and a finite group G having equal character degree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cd(H) = cd(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G ∼= H × A, where A is an abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Huppert’s conjecture has been verified for many cases such as the alternating groups, sporadic groups, and simple groups of Lie type with low rank, but it has yet to be verified for simple groups of Lie type with high rank.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recently, a similar conjecture related to codegrees has been posed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Codegree Version of Huppert’s Conjecture: Let H be a finite nonabelian simple group and G a finite group such that cod(H) = cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G ∼= H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This conjecture appears in the Kourovka Notebook of Unsolved Problems in Group Theory as question 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='79 [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' It has been verified for PSL(2, q), PSL(3, 4), Alt7, J1, 2B2(22f+1) where f ≥ 1, M11, M12, M22, M23 and PSL(3, 3) by [1, 3, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The conjecture has also been verified for PSL(3, q) and PSU(3, q) in [19] and 2G2(q) in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recently, the authors verified the conjecture for all sporadic simple groups in [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this paper, we provide a general proof verifying this conjecture for all alternating groups of degree greater than or equal to 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The methods used may be generalized to simple groups of Lie type, giving promising results for characterizing all simple groups by their codegree sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let An be an alternating group of degree n ≥ 5 and G a finite group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If cod(G) = cod(An), then G ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Throughout the paper, we follow the notation used in Isaacs’ book [16] and the ATLAS of Finite Groups [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2000 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 20C15, 20D06.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Preliminary Results We first introduce some lemmas which will be used later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [21, Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2] Let S be a finite nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then there exists 1S ̸= χ ∈ Irr(S) that extends to Aut(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [17, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='34] Let N be a minimal normal subgroup of G such that N = S1 × · · · × St where Si ∼= S is a nonabelian simple group for each i = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' , t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If χ ∈ Irr(S) extends to Aut(S), then χ × · · · × χ ∈ Irr(N) extends to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [13, Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6] Let G be a finite group and S a finite nonabelian simple group with cod(G) = cod(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then G is a perfect group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' [15] Let G be a finite group and S a finite nonabelian simple group such that cod(S) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then |S| divides |G|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group with N ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [16, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='22], we can define Irr(G/N) = {ˆχ(gN) = χ(g) | χ ∈ Irr(G) and N ⊆ ker(χ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Take any ˆχ ∈ Irr(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By definition, we know that ˆχ(1) = χ(1), so the denominators of cod(ˆχ) and cod(χ) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In addition, ker(ˆχ) ∼= ker(χ)/N, so |ker(χ)| = |N| · |ker(ˆχ)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |G/N : ker(ˆχ)| = |G|/|N| | ker(χ)|/|N| = |G| | ker(χ)|, so cod(ˆχ) = cod(χ) and therefore cod(G/N) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group with normal subgroups N and M such that N ≤ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the Third Isomorphism Theorem, we know that G/M ∼= (G/N)/(M/N) is a quotient of G/N, and by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5, cod(G/M) ⊆ cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let S be a finite nonabelian simple group and G be a nontrivial finite group with cod(G) ⊆ cod(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, |S| < |G| · |Irr(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that for each irreducible character χ ∈ Irr(S), χ(1)2 < |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Because S is simple, if χ is non- trivial, then ker(χ) = 1, so cod(χ) = |S| χ(1) > � |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, since cod(G) ⊆ cod(S), for each irreducible non- trivial character ψ ∈ Irr(G), cod(ψ) > � |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, |G:ker(ψ)| ψ(1) > � |S| which implies that |G| |ker(ψ)|√ |S| > ψ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' So, ψ(1) < |G| √ |S|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then � ψ∈Irr(G) ψ(1)2 < | Irr(G)| |G|2 |S| , and by character theorems, we’ll have |G| < | Irr(G)| |G|2 |S| .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |S| < |G| · |Irr(G)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Main Results We start with some lemmas which limit the simple groups whose codegree set can be contained in the codegree set of an alternating group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be an alternating group of degree m ̸= n, where m, n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose cod(Am) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, from Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |Am| divides |An|, so m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let ax denote the minimal non-trivial codegree of Ax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We show that an−1 < an so that cod(Am) ̸⊆ cod(An) follows immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that irreducible representations of the symmetric group Sn are in one-to-one correspondence with the partitions of n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ be a partition of n and Vλ be the corresponding irreducible representation of Sn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We note that a partition of n can be visualized by a Young diagram and we let hλ(i, j) be the hook length of the (i, j)th square of the Young diagram corresponding to λ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' the number of cells (a, b) of λ such that a = i and b ≥ j or b = j and a ≥ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the hook length formula, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Vλ) = � hλ(i, j) := Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let Uλ be an irreducible constituent of the restriction of Vλ to An, ResSn An Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If λ is not self-conjugate (λ ̸= λ′), then ResSn An Vλ remains irreducible, so Uλ = ResSn An Vλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Uλ) = Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If λ is self- conjugate, then the restriction of Vλ to An splits into two irreducible representations of the same dimension, so dim(Uλ) = 1 2dim(Vλ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' dim(Uλ) = 2Hλ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2 Now, an = min{ n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='/2 dim(Uλ) | Uλ ∈ Irr(An)} = 1 2 min({Hλ | λ ̸= λ′} ∪ {2Hλ | λ = λ′}).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We want to show that an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, assume that an = 1 22Hλ for some λ = λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we can remove a square from λ to give a non-self-conjugate partition µ of n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since Hµ < Hλ < 2Hλ and an−1 ≤ 1 2Hµ, we know an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now assume that an = 1 2Hλ for some λ ̸= λ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 3, we can remove a square from λ to obtain a non-self-conjugate partition µ of n − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since Hµ < Hλ and an−1 ≤ 1 2Hµ, an−1 < an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if m < n, then am < an, contradicting the assumption that cod(Am) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be a sporadic simple group or the Tits group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 5, cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In search of a contradiction, let H be a sporadic simple group or the Tits group such that cod(H) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, we deduce a tight restriction on the order of H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Namely, |H| = |An|/k where 1 ≤ k < |Irr(H)| is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, for each sporadic (or Tits) group H, we can computationally check (using Julia [6]) which alternating groups An satisfy both |H| divides |An| and |An| |H| < |Irr(H)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We find only one possible exception: An = A10 and H = J2 where |A10| |J2| = 3 < 21 = |Irr(J2)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we check that cod(J2) ̸⊆ cod(A10) using the ATLAS [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be a classical simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(H) ̸⊆ cod(An) for all n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' There are 6 families of classical simple groups of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' These are PSL(m + 1, q), Ω(2m + 1, q), PSp(2m, q), O+(2m, q), PSU(m + 1, q), and O−(2m, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='l We prove the lemma in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let k(G) denote the number of conjugacy classes of G, we reproduce [12, Table 2] for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Class Numbers for Classical Groups G k(G) ≤ Comments SL(n, q) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn−1 SU(n, q) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qn−1 Sp(2n, q) 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='8qn q odd Sp(2n, q) 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2qn q even SO(2n + 1, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1qn q odd Ω(2n + 1, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qn q odd SO±(2n, q) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn q odd Ω±(2n, q) 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='8qn q odd O±(2n, q) 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qn q odd SO±(2n, q) 14qn q even O±(2n, q) 15qn q even (1) Let H = PSL(m + 1, q) where q = pk and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From the order formula found in [7], qm(m+1)/2 divides |PSL(m + 1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Legendre’s formula, we know that for any prime p, |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p ≤ p n p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If q = pk, then we have |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|q ≤ q n k(p−1) and thus |An|q ≤ q n k(p−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |PSL(m + 1, q)| divides |An|, so qm(m+1)/2 divides |An|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus m(m+1) 2 ≤ n k(p−1), giving n ≥ m(m+1)k(p−1) 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |An| ≥ ���A m(m+1)k(p−1) 2 ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note that k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) since PSL(m + 1, q) is a quotient of SL(m + 1, q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then from Table 1, we have that |Irr(PSL(m + 1, q))| = k(PSL(m + 1, q)) ≤ k(SL(m + 1, q)) ≤ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 gives |An| < |PSL(m + 1, q)| · |Irr(PSL(m + 1, q))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence |A m(m+1)k(p−1) 2 | < |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now we show that if we consider the left and right sides as functions of m with constants p and k, then asymptotically, the value of |A m(m+1)k(p−1) 2 | grows faster than that of |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We know that the left function behaves asymptotically as (m2)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=', and using the order formula for PSL(m + 1, q), we know that the right function behaves asymptotically as qf(m), where f(m) is a polynomial with degree 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus the left function grows faster than the right function since x!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' >> cx for any constant c when x is large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Similarly, we can prove this result considering the two sides as functions of p and k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 3 Then, we search for the maximum possible value of m which satisfies the inequality given the smallest possible values of p and k, which are 2 and 1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We find that m ≤ 6 and, using a similar process for p and k, that p ≤ 17 and k ≤ 63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we have limited our search to a finite number of groups which we can check in the same way as for the sporadic groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From this, we find a small list of exceptions, listed in Table 2: Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |PSL(m + 1, q)| divides |An| and |An| < |PSL(m + 1, q)| · 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5qm m q n 1 4 5 1 4 6 1 8 7 1 9 6 1 9 7 1 5 5 1 5 6 1 7 7 2 4 8 2 4 9 3 2 8 3 2 9 Now, all of these exceptions can be found in the ATLAS, and it is routine to check that none of these groups satisfy cod(PSL(m + 1, q)) ⊆ cod(An) unless PSL(m + 1, q) ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if PSL(m + 1, q) ̸∼= An, then cod(PSL(m + 1, q) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (2) Let H = Ω(2m+1, q) where q = pk is odd and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Note that when q = 2k is even, Ω(2m+1, q) ∼= PSp(2m, q), which we deal with in the next case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm2 divides |Ω(2m + 1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 1 similarly to above, |Am2k(p−1)| < |Ω(2m+ 1, q)|·7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, we computationally check that we get a contradiction if m > 2, p > 3, or k > 1, so m = 2, p = 3, and k = 1 is the only possibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get the list of exceptions listed in Table 3 after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |Ω(2m + 1, q)| divides |An| and |An| < |Ω(2m + 1, q)| · 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3qm m q n 2 3 9 Again, we check the ATLAS and find that cod(Ω(5, 3)) ̸⊆ cod(A9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (3) Let H = PSp(2m, q) where q = pk and m ≥ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm2 divides |PSp(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since PSp(2m, q) is a quotient of Sp(2m, q), we have k(PSp(2m, q)) ≤ k(Sp(2m, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 1, |Am2k(p−1)| < |PSp(2m, q)| · 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We computationally check that we get a contradiction if m > 4, p > 2, or k > 2, so m = 3 or 4, p = 2, and k = 1 or 2 are the only possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get no exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (4) Let H = O+(2m, q) where q = pk and m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m−1) divides |O+(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Using Table 1, we have that |Am(m−1)k(p−1)| < |O+(2m, q)| · 15qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, we computationally check that we get a contradiction if m > 4, p > 2, or k > 1 so m = 4, p = 2, and k = 1 is the only possibility, and we get no possible exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (5) Let H = PSU(m+1, q) where q = pk and m ≥ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m+1)/2 divides |PSU(m+1, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since PSU(m + 1, q) is a quotient of SU(m + 1, q), we have k(PSU(m + 1, q)) ≤ k(SU(m + 1, q)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 1, |A m(m+1)k(p−1) 2 | < |PSU(m + 1, q)| · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a 4 contradiction if m > 6, p > 7, or k > 42 so m ≤ 6, p ≤ 7, and k ≤ 42 are the only possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We get Table 4 after checking divisibility: Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Exceptions satisfying |PSU(m + 1, q)| divides |An| and |An| < |PSU(m + 1, q)| · 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='26qm m q n 2 3 9 3 2 9 We check the ATLAS to find that cod(PSU(3, 3)) ̸⊆ cod(A9), and we note that PSU(4, 2) ∼= Ω(5, 3), which we have already ruled out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (6) Let H = O−(2m, q) where q = pk and m ≥ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], qm(m−1) divides |O−(2m, q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 1 similarly to above, |Am(m−1)k(p−1)| < |O−(2m, q)| · 15qm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction if m > 5, p > 3, or k > 3 so m ≤ 5, p ≤ 3, and k ≤ 3 are the only possibilities, and we get no possible exceptions after checking divisibility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let H be an exceptional simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then if n ≥ 5, cod(H) ̸⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' There are 10 familes of exceptional simple groups of Lie type (other than the Tits group).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' These are E6(q), E7(q), E8(q), F4(q), G2(q),2 E6(q),3 D4(q),2 B2(q),2 F4(q), and 2G2(q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We prove the lemma in each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, we reproduce [12, Table 1] for reference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Class Numbers for Exceptional Groups G k(G) ≤ Comments 2B2(q) q + 3 q = 22m+1 2G2(q) q + 8 q = 32m+1 G2(q) q2 + 2q + 9 2F4(q) q2 + 4q + 17 q = 22m+1 3D4(q) q4 + q3 + q2 + q + 6 F4(q) q4 + 2q3 + 7q2 + 15q + 31 E6(q) q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60 2E6(q) q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62 E7(q) q7 + q6 + 2q5 + 7q4 + 17q3 + 35q2 + 71q + 103 E8(q) q8 + q7 + 2q6 + 3q5 + 10q4 + 16q3 + 40q2 + 67q + 112 (1) Let H ∼= E6(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From the order formula found in [7], q36 divides |E6(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [5], we know that for any prime p, |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p ≤ p n p−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If q = pk, then we have |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|q ≤ q n k(p−1) and thus |An|q ≤ q n k(p−1) where |An|p is the p-part of An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, |E6(q)| divides |An| so q36 divides |An|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus 36 ≤ n k(p−1) and n ≥ 36k(p − 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |An| ≥ |A36k(p−1)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note from Table 5 that |Irr(E6(q))| = k(E6(q)) ≤ q6 + q5 + 2q4 + 2q3 + 15q2 + 21q + 60.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Applying Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='7 gives |An| < |E6(q)|·|Irr(E6(q))|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, |A36k(p−1)| < |E6(q)|·(q6 +q5 +2q4 + 2q3 + 15q2 + 21q + 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As with the classical Lie type groups, we can computationally find an upper bound on p and k since the left side grows faster in terms of p and k than the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we find that no values of p and k satisfy the inequality, since substituting p = 2 and k = 1 gives |A36| > |E6(2)| · (26 + 25 + 2 · 24 + 2 · 23 + 15 · 22 + 21 · 2 + 60).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, there are no possible values for q and n such that cod(E6(q)) ⊆ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (2) Let H ∼= E7(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q63 divides |E7(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A63k(p−1)| < |E7(q)| · (q7 +q7 +2q5 +7q4 +17q3 +35q2+71q +103).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 5 (3) Let H ∼= E8(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q120 divides |E8(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, we have |A120k(p−1)| < |E8(q)|·(q8+q7+2q6+3q5+10q4+16q3+40q2+67q+112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (4) Let H ∼= F4(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q24 divides |F4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A24k(p−1)| < |F4(q)|·(q4 + 2q3 + 7q2 + 15q + 31).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (5) Let H ∼= G2(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q6 divides |G2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, |A6k(p−1)| < |G2(q)| · (q2 + 2q + 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we find that p = 2, k = 1 satisfies the inequality, but any other values of p and k do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, we note that G2(2) is not simple, so we instead consider its derived subgroup G2(2)′ (which still satisfies the above inequality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We check for exceptions where |G2(2)′| divides |An| and |An| < |G2(2)′| · (22 + 2 · 2 + 9), but there are none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (6) Let H ∼= 2E6(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q36 divides |2E6(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Using Table 5, |A36k(p−1)| < |2E6(q)| · (q6 + q5 + 2q4 + 4q3 + 18q2 + 26q + 62).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for p = 2, k = 1, so there are no possible exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (7) Let H ∼= 3D4(q) where q = pk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q12 divides |3D4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 similarly to above, |A12k(p−1)| < |3D4(q)| · (q4 + q3 + q2 + q + 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we find that p = 2, k = 1 satisfies the inequality, but any other values of p and k do not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As for the sporadic groups, we check for possible exceptions where |3D4(2)| divides |An| and |An| < |3D4(2)| · (24 + 23 + 22 + 2 + 2), but there are none.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (8) Let H ∼= 2B2(q) where q = 22m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q2 divides |2B2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, we have that |A2(2m+1)| < |2B2(q)| · (q + 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we computationally check that we get a contradiction if m > 4, so m must be less than 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, checking the divisibility condition, we get no exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' (9) Let H ∼= 2F4(q) where q = 22m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q12 divides |2F4(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, using Table 5 as above, |A12(2m+1)| < |2F4(q)| · (q2 + 4q + 17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we computationally check that we get a contradiction for m = 1, so there are no exceptions (10) Let H ∼= 2G2(q) where q = 32m+1 and m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [7], q3 divides |2G2(q)|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Table 5, |A3(2m+1)·2| < |2G2(q)| · (q + 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Again, we computationally check that we get a contradiction for m = 1, so there are no exceptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a finite group such that cod(G) = cod(An) where n ≥ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let N be a maximal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, G is perfect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus G/N is a nonabelian simple group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G) = cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4, G/N cannot be an alternating group of degree m ̸= n, a sporadic simple group or the Tits group, a classical simple group of Lie type, or an exceptional simple group of Lie type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ Now we present the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let G be a minimal counterexample and N be a maximal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3, G is perfect, and by Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5, G/N ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In particular, N ̸= 1 as G ̸∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 1: N is a minimal normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose L is a non-trivial normal subgroup of G with L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='6, we have cod(G/N) ⊆ cod(G/L) ⊆ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, cod(G/N) = cod(An) = cod(G) so equality must be obtained in each inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, cod(G/L) = cod(An) which implies that G/L ∼= An since G is a minimal counterexample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction since we also have G/N ∼= An, but L < N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 2: N is the only non-trivial, proper normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Otherwise we assume M is another proper nontrivial normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If N is included in M, then M = N or M = G since G/N is simple, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then N ∩ M = 1 and G = N × M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since M is also a maximal normal subgroup of G, we have N ∼= M ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Choose ψ1 ∈ Irr(N) and ψ2 ∈ Irr(M) such that cod(ψ1) = cod(ψ2) = max(cod(An)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Set χ = ψ1 · ψ2 ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then cod(χ) = (max(cod(An)))2 /∈ cod(G), a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 3: For each non-trivial χ ∈ Irr(G|N) := Irr(G) − Irr(G/N), χ is faithful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 6 We construct Irr(G/N) as the same as Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then it follows by the definition of Irr(G|N) that if χ ∈ Irr(G|N), N ̸≤ ker(χ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus since N is the unique nontrivial, proper, normal subgroup of G, ker(χ) = G or ker(χ) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, ker(χ) = 1 for all nontrivial χ ∈ Irr(G|N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 4: N is an elementary abelian group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose that N is not abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since N is a minimal normal subgroup, by [10, Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='3A (iii)], N = Sn where S is a nonabelian simple group and n ∈ Z+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Lemmas 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='2, there is a non- trivial character χ ∈ Irr(N) which extends to some ψ ∈ Irr(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, ker(ψ) = 1 by Step 3, so cod(ψ) = |G|/ψ(1) = |G/N| · |N|/χ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, by assumption, we have that cod(G) = cod(An) = cod(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, cod(ψ) ∈ cod(G) = cod(G/N), so cod(ψ) = |G/N|/φ(1) for some φ ∈ Irr(G/N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, |G/N| is divisible by cod(ψ) which contradicts the fact that cod(ψ) = |G/N| · |N|/χ(1), as χ(1) ̸= |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus N must be abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now to show that N is elementary abelian, let a prime p divide |N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then N has a p-Sylow subgroup K, and K is the unique p-Sylow subgroup of N since N is abelian, so K is characteristic in N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, K is a normal subgroup of G, so K = N as N is minimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |N| = pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, take the subgroup N p = {np | n ∈ N} of N, which is proper by Cauchy’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since N p is characteristic in N, it must be normal in G, so N p is trivial by the uniqueness of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus every element of N has order p, and N is elementary abelian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 5: CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First note that since N is normal, CG(N) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Additionally, since N is abelian by Step 4, N ≤ CG(N).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the maximality of N, we must have CG(N) = N or CG(N) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If CG(N) = N, we are done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If not, then CG(N) = G, so N must be in the center of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then since N is the unique minimal normal subgroup of G by Step 2, we must have that |N| is prime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If not, there always exists a proper non-trivial subgroup K of N, and K is normal since it is contained in Z(G), contradicting the minimality of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, since G is perfect, we have that Z(G) = N, and N is isomorphic to a subgroup of the Schur multiplier of G/N [16, Corollary 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, we note that it is well-known that for n > 7, the Schur multiplier of An is Z2, so G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From [20], 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An always has a character degree of order 2⌊(n−2)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let χ be such an irreducible character of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An with χ(1) = 2⌊(n−2)/2⌋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Recall that by Step 2, there is only one non-trivial proper normal subgroup of G ∼= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In particular N ∼= Z2 is the only non-trivial proper normal subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus |ker(χ)| = 1 or 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we have cod(χ) = |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An:ker(χ)| χ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' If |ker(χ)| = 1, then cod(χ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2⌊(n−2)/2⌋ , and if |ker(χ)| = 2, then cod(χ) = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='/2 2⌊(n−2)/2⌋ = n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 2⌊n/2⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In either case, for any prime p ̸= 2, | cod(χ)|p = |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|p = |An|p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since cod(G) = cod(An), we know that cod(χ) ∈ cod(An).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, there is a character degree of An which is a power of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, from [20], we know that for n > 7, An only has a character degree equal to a power of 2 when n = 2d + 1 for some positive integer d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, 2d = n − 1 ∈ cd(An) so we need |An| n−1 = |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An| 2⌊(n−2)/2⌋ or |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='An| 2⌊n/2⌋ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, 1 n−1 = 2 2⌊(n−2)/2⌋ = 1 2⌊(n−2)/2⌋−1 or 1 2⌊n/2⌋−1 so n − 1 = 2⌊(n−2)/2⌋−1 or 2⌊n/2⌋−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, the only integer solution to either of these equations occurs when n = 9 and 9 − 1 = 8 = 23 = 2⌊9/2⌋−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, we check the ATLAS [9] to find that the codegree sets of A9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='A9 do not have the same order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction, so CG(N) = N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 6: Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ), the set of irreducible constituents of λIG(λ), where IG(λ) is the inertia group of λ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Also, ϑ(1) divides |IG(λ)/N|, and |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Lastly, IG(λ) < G, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' λ is not G-invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ be a non-trivial character in Irr(N) and ϑ ∈ Irr(IG(λ)|λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let χ be an irreducible constituent of ϑG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By [16, Corollary 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='4], we know χ ∈ Irr(G), and by [16, Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1], we have χ(1) = |G| |IG(λ)| · ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, we know tat ker(χ) = 1 by Step 2, and thus cod(χ) = |G| χ(1) = |IG(λ)| ϑ(1) , so |IG(λ)| ϑ(1) ∈ cod(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, since N is abelian, λ(1) = 1, so we have ϑ(1) = ϑ(1)/λ(1) which divides |IG(λ)| |N| , so |N| divides |IG(λ)| ϑ(1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Moreover, we know that cod(G) = cod(G/N), and all elements in cod(G/N) divide |G/N|, so |N| divides |G/N|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Next, we want to show IG(λ) is a proper subgroup of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' To reach a contradiction, assume IG(λ) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then ker(λ) ⊴ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Step 2, we know ker(λ) = 1, and from Step 4, we know N is a cyclic group of prime 7 order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus by the Normalizer-Centralizer theorem, we have G/N = NG(N)/CG(N) ≤ Aut(N) so G/N is abelian, a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Step 7: Final contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' From Step 4, N is an elementary abelian group of order pm for some prime p and integer m ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By the Normalizer-Centralizer theorem, An ∼= G/N = NG(N)/CG(N) ≤ Aut(N) and m > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Note that in general, Aut(N) = GL(m, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Step 6, |N| divides |G/N|, so we know that |N| = pm divides |An| and G/N ∼= An ≲ GL(m, p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We prove by contradiction that this cannot occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, we claim that if pm divides |An| and An ≲ (GL(m, p), then p must equal 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' To show this, we note that for p > 2, by [5], we have that if pm divides |An|, then m < n 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 of [24] shows that if n > 6, the minimal faithful degree of a modular representation of An over a field of characteristic p is at least n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since embedding An as a subgroup of GL(m, p) is equivalent to giving a faithful representation of degree m over a field of characteristic p, we have that m ≥ n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This is a contradiction since n 2 > n − 2 implies n < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, let p = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, from [5], we obtain |n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='|2 ≤ 2n−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, if 2m divides |An|, then m ≤ |An|2 ≤ 2n−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='1 of [23] shows that if n > 8, then the minimal faithful degree of a modular representation of An over a field of characteristic 2 is at least n − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, we must have m ≥ n − 2, so m = |An|2 = 2n−2 is the only option.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Let λ ∈ Irr(N), ϑ ∈ Irr(IG(λ)|λ), and T := IG(λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then 1 < |G : T | < |N| = 2n−2 for |G : T | is the number of all conjugates of λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' By Step 5, we know that |T | ϑ(1) ∈ cod(G) and moreover that |N| divides |T | ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Since |N|2 = |An|2 and cod(G) = cod(An), we know that ��� |T | ϑ(1) ��� 2 = |N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus ��� |T/N| ϑ(1) ��� 2 = 1 so the 2-parts of |T/N| and ϑ(1) are equal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus for every ϑ ∈ Irr(T | λ), we have |ϑ(1)|2 = |T/N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, |T/N| = � ϑ∈Irr(T |λ) ϑ(1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Hence, if |ϑ(1)|2 = 2k ≥ 2 for every ϑ ∈ Irr(T | λ), we would have |T/N|2 = 22k contradicting the fact that |ϑ(1)|2 = |T/N|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Therefore, |T/N|2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, since |G/N|2 ≥ |N|2 = 2n−2, we have |G : T |2 = |G/N : T/N|2 ≥ 2n−2, so |G : T | ≥ 2n−2 = |N|, which is a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We have one final exception to consider: n = 8, p = 2, and m = 4, 5 or 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In this case, A8 ∼= GL(4, 2) and 26 divides |A8|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, cod(A8) = {1, 26·32·5, 25·32·5, 24·32·7, 26·3·5, 24·32·5, 26·32, 26·7, 23·32·5, 32·5·7, 25·32} from [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' We will look at each possibility for m in turn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' First, let m = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then we have G/N ∼= A8 ∼= GL(4, 2), N = (Z2)4 so G is an extension of GL(4, 2) by N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Suppose first that this extension is split and G is a semidirect product.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' This semidirect product is defined by a homomorphism φ : GL(4, 2) → Aut((Z2)4) ∼= GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, since GL(4, 2) is simple, ker(φ) = 1 or GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In the first case, we have the trivial direct product, so there are at least two copies of GL(4, 2) as normal subgroups of G, which contradicts Step 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' In the second case, φ is some automorphism of GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Here, we can check using GAP that any such φ creates a semidirect product GL(4, 2) ⋊φ (Z2)4 which does not have the same codegree set as A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, suppose that the extension is non-split.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Then, [4] gives that there is a unique non-split extension 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='GL(4, 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' However, we find using GAP that it doesn’t have the same codegree set as A8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Second, let m = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' As above, |G : T | < |N| = 25 and |T | ϑ(1) ∈ cod(G) such that 25 divides |T | ϑ(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Further, | |T/N| ϑ(1) |2 ≤ 2 so |T/N|2 ≤ 4 and |G/N : T/N|2 ≥ 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus, we have 16 divides |G/N : T/N| and |G/N : T/N| < 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' But we check the index of all subgroups of G/N ∼= A8 using GAP and find that none of them satisfy these two properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Finally, let m = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Now, |N|2 = |A8|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' For this case the same argument as above for general An holds, and we reach a contradiction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Thus we find that every |N| = pm produces a contradiction, so N = 1 and G ∼= An.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Acknowledgements This research was conducted under NSF-REU grant DMS-1757233, DMS-2150205 and NSA grant H98230- 21-1-0333, H98230-22-1-0022 by Dolorfino, Martin, Slonim, and Sun during the Summer of 2022 under the supervision of Yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The authors gratefully acknowledge the financial support of NSF and NSA, and also thank Texas State University for providing a great working environment and support.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Yang was also partially supported by grants from the Simons Foundation (#499532, #918096, to YY).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' The authors would also like to thank Prof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Richard Stanley for his help.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' 8 References [1] N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Ahanjideh, Nondivisibility among irreducible character co-degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Bull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content=' Mallory Dolorfino, Kalamazoo College, Kalamazoo, Michigan, USA, mallory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='dolorfino19@kzoo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Luke Martin, Gonzaga University, Spokane, Washington, USA, lwmartin2019@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='com Zachary Slonim, University of California, Berkeley, Berkeley, California, USA, zachslonim@berkeley.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Yuxuan Sun, Haverford College, Haverford, Pennsylvania, USA, ysun1@haverford.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu Yong Yang, Texas State University, San Marcos, Texas, USA, yang@txstate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'} +page_content='edu 9' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ENE0T4oBgHgl3EQfywJf/content/2301.02663v1.pdf'}