Complex group algebras of almost simple unitary groups.

The aim of this article is to contribute to a question of R. Brauer that "when do non-isomorphic groups have isomorphic complex group algebras?" Let H and G be finite groups where PSU n (q) ≤ G ≤ PGU n (q) , and let X 1 (H) denote the first column of the complex character table of H. In this article, we show that if X 1 (H) = X 1 (G) , then H ≅ G provided that q + 1 divides neither n nor n – 1. Consequently, it is shown that G is uniquely determined by the structure of its complex group algebra. This in particular extends a recent result of Bessenrodt et al. [Algebra Number Theory 9 (2015), 601–628] to the almost simple groups of arbitrary rank. [ABSTRACT FROM AUTHOR]