RECOGNITION OF FINITE QUASI-SIMPLE GROUPS BY THE DEGREES OF THEIR IRREDUCIBLE REP- RESENTATIONS HUNG NGOC NGUYEN∗ and HUNG P. TONG-VIETy ∗Department of Mathematics, The University of Akron, Akron, Ohio 44325, United States Email: [email protected] ySchool of Mathematics, Statistics and Computer Science, University of KwaZulu- Natal, Scottsville 3209, South Africa Email: [email protected] Abstract We present some recent advances in the study of the problem of recognizing finite groups by the degrees of their irreducible complex representations. We especially focus on simple groups and more generally quasi-simple groups. 1 Introduction Representation theory of finite groups was originally developed to analyze groups in terms of linear transformations or matrices. A representation of degree n (where n is a positive integer) over a field F of a group is a way to represent elements in the group by n × n invertible matrices with entries in F in such a way that the rule of group operation corresponds to matrix multiplication. Degree certainly is the most important information in a rep- resentation and therefore, the degrees of irreducible representations are those of the key tools to study the structure of finite groups. This is an expository paper in which we survey some recent advances on the problem of recognizing finite groups by the degrees of their (complex) representations, especially for simple groups and more generally quasi-simple groups. For a finite group G, we denote the set of degrees of irreducible representations of G by cd(G) and call it simply as degree set of G. The multiplicity of each degree is the number of irreducible representations of that degree, and when these numbers are taken into account, we will similarly have the degree multiset of G, denoted by cd∗(G). A fundamental question in group representation theory is whether one Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 2 can recover a group or some of its properties from the degrees of its irre- ducible representations. In the late 1980s, I. M. Isaacs [18] proved that if cd∗(G) = cd∗(H) and p is a prime, then G has a normal p-complement if and only if H has a normal p-complement, and therefore the nilpotency of a group is determined by its degree multiset. Later, T. Hawkes [14] provided a counterexample showing that the same assertion does not hold for super- solvability. It is still unknown whether the solvability of a finite group is determined by its degree multiset, see [28, Problem 11.8]. In his famous list of problems in representation theory of finite groups [9], R. Brauer asked: when do non-isomorphic groups have isomorphic complex group algebras? (see [9, Problem 2]). The complex group algebra of a finite group G, denoted by CG, is isomorphic to a direct sum of matrix algebras over C whose dimensions are exactly the degrees of irreducible representations of G. Therefore, Brauer's question leads to the following: Problem 1.1 Given a finite group G, determine all finite groups (up to iso- morphism) having the same degree multiset with G. Though a complete solution to this problem seems out of reach based on the present knowledge of group representation theory, one may hope to obtain a partial solution. Problem 1.1 is easy for abelian groups but difficult for solvable groups in general. This is due to the fact that the connection between a solvable group and its degree multiset is rather loose in the sense that there are often several non-isomorphic groups having the same degree multiset with a given solvable group. In contrast to solvable groups, simple groups or groups `close' to simple seem to have a stronger connection with their representation degrees. In Sections2, we sketch some main ideas in the solution of Problem 1.1 for simple groups, mainly due to the second author. In Sections3, we discuss the problem for other groups close to simple such as quasi-simple groups and almost simple groups. Especially, we will present a method to approach the conjecture that every quasi-simple group is determined uniquely up to isomorphism by its degree multiset. We have seen a tight relation between a quasi-simple group and its multiset of irreducible representation degrees. In the late 1990s, B. Huppert proposed that the connection should be tighter, at least for non-abelian simple groups. In fact, he conjectured in [15] that if G is a finite group and S is a finite non-abelian simple group such that the degree sets of G and S are the same, then G is isomorphic to the direct product of S and an abelian group. To give some evidence, Huppert himself verified the conjecture on a case-by-case basis Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 3 for many simple groups, including the Suzuki groups, many of the sporadic simple groups, and a few of the simple groups of Lie type. Recently, there has been substantial progress on verifying Huppert conjecture by the authors and their collaborators, especially for various families of simple groups of Lie type of small rank. This is discussed in Section4. Recent success on Problem 1.1 for quasi-simple groups suggests that Hup- pert conjecture might be extended from non-abelian simple groups to quasi- simple groups. The following conjecture has been recently proposed in [37] and will be discussed in Section5. Conjecture 1.2 Let G be a finite group and H a finite quasi-simple group. If cd(G) = cd(H), then G ∼= H ◦ A, a central product of H with an abelian group A. In other words, every finite quasi-simple group is determined up to an abelian central product factor by its degree set. In the last section, we report some recent results concerning the recog- nition of non-abelian simple groups by using the multiplicity of irreducible representation degrees. Notice that if the complex group algebra CG of some finite group G is given, then we know both cd(G) and the multiplicity pattern mp(G) (defined in Section6). It is shown in [49] that several families of non- abelian simple groups are uniquely determined by the multiplicity patterns and we conjecture that every non-abelian simple group is uniquely determined by its multiplicity pattern. Perhaps the best way to describe complex representations (and indeed mod- ular representations as well) is through characters. This is due to the fact that a complex representation of a finite group is determined (up to equiva- lence) by its character. The character afforded by a group representation is a function on the group which associates to each group element the trace of the corresponding matrix and therefore it carries the essential information about the representation in a more condensed form. The degree of a character is exactly the degree of the representation affording it. This explains why we have used the notations cd(G) and cd∗(G) (c.d. stands for character degree) for the set and multiset of irreducible representation degrees. Throughout the paper, we will go back and forth between representations and characters, depending on which one is more convenient. 2 Simple Groups and Their Degree Multisets If G is any finite abelian group of order n, then CG is isomorphic to a direct sum of n copies of C so that the degree multisets of any two abelian groups Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 4 having the same order are equal. For p-groups or more generally nilpotent groups, the probability that two groups have equal degree multisets is also fairly `high'. For instance, Huppert pointed out in [15] that among 2328 groups of order 27, there are only 30 different degree multisets. In [13], Dade even managed to construct two non-isomorphic metabelian groups G and H with isomorphic group algebras FG and FH over an arbitrary field F. We now turn our attention to finite non-abelian simple groups. As men- tioned above, simple groups seem to have a stronger connection with their representation degrees. In [46, 47, 48], the second author has succeeded in proving that, if G is a finite group and S is a finite non-abelian simple group such that cd∗(G) = cd∗(S), then G ∼= S. This substantially improves a classi- ∼ ∼ cal result of W. Kimmerle in [19] where it was proved that G = S if FG = FS for every field F. ∗ ∗ Obverse that knowing cd (G) is equivalent to knowing CG and cd (G) is just the first column of the ordinary character table of G: There are sev- eral papers in the literature devoted to characterizing the non-abelian simple groups by their character tables (see, for example, [23, 38, 39, 40]). Upon the completion of the classification, it is easy to see that all non-abelian simple groups are uniquely determined by their character tables. In fact, as the nor- mality of subgroups of a group can be detected from the character table, if finite groups G and S have the same character table, where S is non-abelian simple, then G is also non-abelian simple, and furthermore jGj = jSj: Now by applying Artin's Theorem [20, Theorem 5.1], we have fG; Sg = fPSL4(2); PSL3(4)g or fG; Sg = fPSp2n(q); Ω2n+1(q)g; where n ≥ 3 and q is an odd prime power. For these exceptions, we can easily check that they have distinct character tables. Thus, we have proved that if a finite group G and a finite non-abelian simple group S have the same character table, then G ∼= S: The main result in this section, stated below, gives a new characterization of finite non-abelian simple groups by using the first column of their ordinary character tables or equivalently, by their complex group algebras.