Composition Factors from the Group Ring and Artin's Theorem on Orders of Simple Groups

Abstract

The integral group ring of a finite group determines the isomorphism type of the chief factors of the group. Two proofs are given, one of which applies Cameron's and Teague's generalisation of Artin's theorem on the orders of finite simple groups to the orders of characteristically simple groups. The generalisation states that a direct power of a finite simple group is determined by its order with the same two types of exception which Artin found. Its proof, given here in detail, adapts and makes explicit certain functions of a natural number variable which Artin used implicitly. These functions contribute to the argument through a series of tables which supply their values for the orders of finite simple groups.

Simple groups are characterized by their non-commuting graphs

“…The following result is a slight extension of a theorem proved by Emil Artin in [3] for most simple groups of Lie type 1 and later extended in [16] to all remaining such groups by Kimmerle, et al Our reason for focusing on inner-diagonal automorphisms will become apparent in the later sections.…”

“…Later, in [16], Kimmerle et al treat the remaining classes of simple groups: 2 B 2 , 3 D 4 , 2 G 2 , 2 F 4 , and 2 E 6 . However, among these latter classes, only the groups 2 E 6 .q/ admit outer diagonal automorphisms (of order d D .3; q C 1/).…”

“…A similar argument applies to ¹ 2 E 6 .q/; U 9 .q/º, since j 2 E 6 .q/j has Zsigmondy factor q 12 1, while jU 9 .q/j does not. Finally, we rule out ¹E 8 .q/; U 16 .q/º by virtue of the fact that jU 16 .q/j has Zsigmondy factor q 16 1, while jE 8 .q/j does not. This completes the proof of our main theorem.…”

On the Generating Graph of a Simple Group

“…From [9], we see that λ(|T |) < 0.49 if T is not a classical group and λ(|T |) < 1/2−1/(c 1 · ) if T is a classical group where c 1 > 2 is some absolute constant and is the Lie rank of T .…”

Grupos finitos e quebra de simetria no código genético

“…No caso dos grupos finitos simples não-abelianos, temos o teorema de Artin sobre as ordens dos grupos finitos simples [5,6,97] que afirma que para dois grupos finitos simples não-abelianos de mesma ordem ocorre exatamente uma das seguintes possibilidades 1) eles são isomorfos;…”