Characterizing Finite Quasisimple Groups by 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 (Nguyen, Israel J Math, 2013) that a quasisimple classical group L is uniquely determined up to isomorphism by the structure of ${{\mathbb C}} L$ , the complex group algebra of L, when L/Z(L) is not isomorphic to PSL₃(4) or PSU₄(3). In this paper, we establish the 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 Tong-Viet (Monatsh Math 166(3–4):559–577, 2012, Algebr Represent Theor 15:379–389, 2012), we 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.