CHARACTERIZING FINITE QUASISIMPLE GROUPS BY THEIR COMPLEX GROUP ALGEBRAS HUNG NGOC NGUYEN AND HUNG P. TONG-VIET Abstract. 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 [12] that a quasisimple classical group L is uniquely determined up to isomorphism by the structure of CL, 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 [14, 16], 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 algebras. 1. Introduction For a finite group G, let CG denote the complex group algebra of G over the field of complex numbers; i.e. CG is the set of formal sums X f agg j ag 2 Cg g2G equipped with natural rules for addition, multiplication, and scalar multiplication. As it is well known that CG is isomorphic to the direct sum of matrix algebras over C whose dimensions are exactly the degrees of irreducible complex representations of G, the study of complex group algebras and the relation to their base groups plays an important role in group representation theory. A fundamental question in representation theory of finite groups is whether one can recover a group or some of its properties from its complex group algebra. For instance, it was proved by M. Isaacs [6] that if two groups have isomorphic complex group algebras and one of them is nilpotent, then the other one is also nilpotent. The corresponding problem for solvability seems very difficult and is still open (see [10, Problem 11.8]). Date: May 2, 2012. 2010 Mathematics Subject Classification. Primary 20C33, 20C15. Key words and phrases. Complex group algebras, quasisimple groups, exceptional groups, groups of Lie type, sporadic groups. 1 2 H.N. NGUYEN AND H.P. TONG-VIET In general, a solvable group is not determined uniquely up to isomorphism by its complex group algebra. Especially for abelian groups and p-groups, the probability that two nonisomorphic groups have the same complex group algebra is fairly \high". On the other hand, it is believed that nonabelian simple groups or more generally quasisimple groups have a stronger connection with their complex group algebras. Conjecture 1.1. Every finite quasisimple group is determined uniquely up to iso- morphism by the structure of its complex group algebra. We recall that 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 perfect central cover or simply a cover of L=Z(L). In [14, 15, 16], the second author has confirmed the conjecture for all nonabelian simple groups. It has been proved recently by the first author [12] that a quasisimple classical group L is uniquely determined up to isomorphism by the structure of CL if 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 moreover for all covers with nontrivial center of simple groups of exceptional Lie type and sporadic groups. In particular, we prove the following in Section3. Theorem 1.2. Let G be a finite group and let L be the Schur cover of E6(q), E7(q), 2 ∼ ∼ or E6(q), where q is a prime power. If CG = CL then G = L. The exceptional covers of simple groups of exceptional Lie type such as 2 · F4(2), 2 2 2 3·G2(3), 2·G2(4), f2; 2 g·Sz(8), and f2; 3; 2 ; 6g· E6(2) are treated in Section4. The following theorem, which confirms the conjecture for all covers of sporadic groups, will be proved in Section5. Theorem 1.3. Let G be a finite group and let L be the a cover of a sporadic simple ∼ ∼ group. If CG = CL then G = L. As a consequence of our results, we obtain: Corollary 1.4. Every finite quasisimple group except possibly covers of the alternat- ing groups is uniquely determined up to isomorphism by the structure of its complex group algebra. Proof. This is a consequence of Theorems 1.2, 1.3, 4.1 and 6.1, and the main results of [12, 14, 16]. We now describe briefly some ideas in the proofs. Let cd(G) denote the set of all irreducible character degrees of G. If G is a finite group and L is a quasisimple group having the same complex group algebra then certainly they have the same set of character degrees. That is, cd(G) = cd(L). Moreover, as L is quasisimple, it has a unique linear character and and hence G also has a unique linear character. It follows that G is perfect and therefore if M is a maximal normal subgroup of G then G=M COMPLEX GROUP ALGEBRAS OF QUASISIMPLE GROUPS 3 is a nonabelian simple group. Let Schur(L=Z(L)) denote the Schur cover of L=Z(L). Since every cover of L=Z(L) is a quotient of Schur(L=Z(L)), we have cd(G=M) ⊆ cd(G) = cd(L) ⊆ cd(Schur(L=Z(L))): We will use this condition together with some others if necessary to force two non- abelian simple groups G=M and L=Z(L) to be isomorphic. This basically eliminates the involvement of all nonabelian simple groups other than L=Z(L) in the structure of G. ∼ Note that as CG = CL, we also have jGj = jLj. It then follows that jMj = jZ(L)j ≤ jMult(L=Z(L))j, where Mult(L=Z(L)) is the Schur multiplier of L=Z(L). Using this together with the facts that G=M ∼= L=Z(L) and cd(G) ⊆ cd(Schur(L=Z(L))), we prove in Section2 that G is uniquely determined by L=Z(L) and the order of G whenever L is not a cover of PSL3(4) or PSU4(3). As the Schur multipliers of these two simple groups are large, we need a closer analysis of the character degrees of their covers and this will be done in Section6. 2. Preliminaries We first define some notation that will be used throughout the paper. For a finite group G, let d(G) and b(G) denote the smallest and largest, respectively, degree of nontrivial irreducible characters of G. The Schur multiplier of G is denoted by Mult(G). If G is perfect, it is well-known that G has a unique Schur cover, which we denote by Schur(G): We write Irr(G) and cd(G) for the set of all complex irreducible characters of G and the set of all irreducible character degrees of G; respectively. By π(G) we mean the set of all prime divisors of the order of G. We will prove in this section some useful facts that will be needed in the proofs of the main theorems. We first recall two important lemmas. The first one is due to A. Moret´oand the second one is due to M. Bianchi, D. Chillag, M.L. Lewis, and E. Pacifici. Lemma 2.1 ([11], Lemma 4.2). Let S be a nonabelian simple group. Then there exists a non-principal irreducible character of S that extends to Aut(S). Lemma 2.2 ([1], Lemma 5). Let N = S × · · · × S, a direct product of k copies of a nonabelian simple group S, be a minimal normal subgroup of G. If χ 2 Irr(S) extends to Aut(S), then χk extends to G. The following is a key result of [12]. Lemma 2.3 ([12], Lemma 7.4). Let S be a nonabelian simple group. Let G be a perfect group and M ¡ G such that G=M ∼= S and jMj ≤ jMult(S)j. Then, for every nonnegative integer i, G=M (i) is isomorphic to a quotient of Schur(S). Using this lemma, we can prove an extension of [12, Lemma 7.5]. 4 H.N. NGUYEN AND H.P. TONG-VIET Lemma 2.4. Let S be a nonabelian simple group different from an alternating group of degree greater than 13. Let G be a perfect group and M ¡ G such that G=M ∼= S, jMj ≤ jMult(S)j, and cd(G) ⊆ cd(Schur(S)). Then G is isomorphic to a quotient of Schur(S). Proof. The proof is very similar to that of [12, Lemma 7.5]. Indeed, the lemma is already proved there for simple groups of Lie type. Therefore, we can assume that S is a sporadic group or an alternating group of degree less than 14. If M is solvable then we are done by Lemma 2.3. So it remains to consider the case when M is nonsolvable. Let M (i) denote the ith derived subgroup of M. Since M is nonsolvable, there is a nonnegative integer i such that M (i) = M (i+1) > 1: Let N ≤ M (i) be a normal subgroup of G so that M (i)=N ∼= T k, a direct product of copies of a nonabelian simple group T . By Lemma 2.1, the group T has a non-principal irreducible character ' that extends to Aut(T ).