Isotypies for the quasisimple groups with exceptional Schur multiplier

Let [Formula: see text] be a block with abelian defect group [Formula: see text] of a quasisimple group [Formula: see text], such that [Formula: see text] has exceptional Schur multiplier. We show that, [Formula: see text] is isotypic to its Brauer correspondent in [Formula: see text] in the sense of Broué. The proof uses methods from a previous paper [B. Sambale, Broué’s isotypy conjecture for the sporadic groups and their covers and automorphism groups, Internat. J. Algebra Comput. 25 (2015) 951–976], and relies ultimately on computer calculations. Moreover, we verify the Alperin–McKay conjecture for all blocks of [Formula: see text].

Recognition of PSL(2,p) ×PSL(2,p) by its complex group algebra

In [H. P. Tong-Viet, Simple classical groups of Lie type are determined by their character degrees, J. Algebra 357 (2012) 61–68] the following question arose: Question. Which groups can be uniquely determined by the structure of their complex group algebras? It is proved that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of [Formula: see text], the complex group algebra of [Formula: see text]. One of the next natural groups to be considered are the characteristically simple groups. In this paper, as the first step in this investigation we prove that if [Formula: see text] is an odd prime number, then [Formula: see text] is uniquely determined by the structure of its complex group algebra.