Characterizing finite quasisimple groups by their complex group algebras.
A fundamental question in the representation theory of finite groups is whether one can recover a group or some of its properties from its complex group algebra. It is believed that nonabelian simple groups or more generally quasisimple groups have a strong connection with their complex group algebras. A finite group \(L\) is said to be quasisimple if \(L\) is perfect and \(L/Z(L)\) is nonabelian simple, in which case we also say that \(L\) is a cover of \(L/Z(L)\). It has been proved recently [\textit{H. N. Nguyen}, Isr. J. Math. 195, Part B, 973-998 (2013; Zbl 1294.20004)] that a quasisimple classical group \(L\) is uniquely determined up to isomorphism by the structure of \(\mathbb CL\), the complex group algebra of \(L\), when \(L/Z(L)\) is not isomorphic to \(\text{PSL}_3(4)\) or \(\text{PSU}_4(3)\). In this paper, the authors establish a similar result for these two open cases and also for covers with nontrivial center of simple groups of exceptional Lie type and sporadic groups. Together with the main results of \textit{H. P. Tong-Viet} [Monatsh. Math. 166, No. 3-4, 559-577 (2012; Zbl 1255.20006); Algebr. Represent. Theory 15, No. 2, 379-389 (2012; Zbl 1252.20005)], they obtain that every quasisimple group except covers of the alternating groups is uniquely determined up to isomorphism by the structure of its complex group algebra.
scientific article; zbMATH DE number 6306467
scientific article