Prof. Ali Iranmanesh, Ph.D.

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Influence of Character Degrees on the Structure of Nearly Simple Groups

A fundamental question in representation theory of finite groups is the extent to which complex group algebra or character degree set of a finite group determines the group or some of its properties.

It is known that, in general, the complex group algebra or the character degree set of a finite solvable group does not determine the group structure up to isomorphism. In contrast to solvable groups, the non-abelian (nearly) simple groups seems to have a stronger relation to their complex group algebras or their set of character degrees. Indeed, it has been recently shown in [1] that finite quasi-simple groups are determined uniquely (up to isomorphism) by the structure of their complex group algebras. Furthermore, a celebrated conjecture of Huppert [2] states that finite non-abelian simple groups are uniquely determined up to an abelian direct factor by the set of their character degrees.

In this article, we will survey on recent improvements of the above results including our recent project aimed at extending the above results to almost simple groups. In particular, we will discuss the following results.

Theorem A. ([3], [4]) Let n \geq 2 and PSL_n^{\epsilon}(q) \leq G \leq PGL_n^{\epsilon} (q) be an almost simple group where q-\epsilon1 is not a divisor of n or n-1. Then G is determined up to isomorphism by the structure of its complex group algebra (here, \epsilon=+ if G is of linear type, and \epsilon = - if G is of unitary type).

We have also proposed an extension of Huppert’s conjecture from non-abelian simple groups to almost simple groups of Lie type.

Conjecture B. Let G be a finite group and H be an almost simple group of Lie type with cd(G) = cd(H). Then G/H \cong H for some abelian normal subgroup A of G.

Inview of Huppert’s conjecture, we show that G is not necessarily the direct product of H and A, and also the converse implication does not necessarily hold for almost simple groups. Furthermore, I will explain our recent work on verifying Conjecture B for some almost simple groups of Lie type of low ranks [5], and also on verifying the Huppert original conjecture for the family of projective special linear groups PSL_5 (q) .

References
[1] Bessenrodt, C., Nguyen, H. N., Olsson, J. B., and Tong-Viet, H. P: Complex group algebras of the double covers of the symmetric and alternating groups. Algebra Number Theory 9 (2015), pp. 601–628.

[2] Huppert, B. Some simple groups which are determined by the set of their character degrees. I. Illinois J. Math., 44 (2000) 828–842.

[3] Shirjian, F and Iranmanesh A: Complex group algebras of almost simple groups with socle PSL n (q). Comm. Algebra 46(2) (2018), pp. 552–573.

[4] Shirjian, F, Iranmanesh, A and Shafiei, F: Complex group algebras of almost simple unitary groups. submitted for publication.

[5] ShirjianA,andIranmanesh, A.ExtendingHuppert?sconjecturetoalmostsimple groups of Lie type, submitted for publication.

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