On groups with the same character degrees as almost simple groups with socle sporadic simple groups

TL;DR

This paper investigates the structure of finite groups with the same set of irreducible character degrees as almost simple groups with sporadic simple socles, revealing their close relationship to the socle and highlighting limitations of Huppert's conjecture.

Contribution

It proves that such groups are closely related to the socle and provides examples showing the conjecture does not extend to almost simple groups.

Findings

G's derived subgroup equals the socle of H

Existence of an abelian subgroup A with G/A isomorphic to H

Counterexamples to extending Huppert's conjecture to almost simple groups

Abstract

Let G be a finite group and cd(G) denote the set of complex irreducible character degrees of G. In this paper, we prove that if G is a finite group and H is an almost simple group whose socle is a sporadic simple group H0 such that cd(G) = cd(H), then G' = H0 and there exists an abelian subgroup A of G such that G/A is isomorphic to H. In view of Huppert's conjecture (2000), we also provide some examples to show that G is not necessarily a direct product of A and H, and hence we cannot extend this conjecture to almost simple groups.