Recognition of Finite Quasi-Simple Groups by the Degrees of Their Irreducible Rep- Resentations

RECOGNITION OF FINITE QUASI-SIMPLE GROUPS BY THE DEGREES OF THEIR IRREDUCIBLE REP- RESENTATIONS

HUNG NGOC NGUYEN∗ and HUNG P. TONG-VIET†

∗Department of Mathematics, The University of Akron, Akron, Ohio 44325, United States Email: [email protected] †School 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 representation 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 irreducible 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 isomorphism) 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 recognition 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 modular 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 mentioned 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 several 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 |G| = |S|. Now by applying Artin’s Theorem [20, Theorem 5.1], we have

{G, S} = {PSL4(2), PSL3(4)} or

{G, S} = {PSp2n(q), Ω2n+1(q)}, 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.

Theorem 2.1 ([46, 47, 48]) If G is a ﬁnite group and S is a ﬁnite non- abelian simple group such that cd∗(G) = cd∗(S), then G ∼= S. In other words, every non-abelian simple group is determined uniquely up to isomorphism by its degree multiset. Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 5

The result above is a weaker version of Huppert conjecture proposed by B. Huppert in the late 1990s, which will be considered in Section4. In- deed, if H is any non-abelian simple group and G is a finite group such that cd∗(G) = cd∗(S), then cd(G) = cd(S) and |G| = |S|. In particular, if Huppert conjecture holds for the simple group S, then since cd(G) = cd(S), we deduce that G ∼= S × A, for some abelian group A. By comparing the orders, we deduce that G ∼= S as wanted. As expected, we have made use of the classification of finite simple groups in the proof of Theorem 2.1. We know that every non-abelian simple group is a sporadic simple groups, an alternating group of degree at least 5, or a finite simple group of Lie type. The latter class can be divided into the finite simple classical groups and the finite simple exceptional groups of Lie type. We first record some easy properties of groups G and S under the hypothesis that cd∗(G) = cd∗(S), where S is non-abelian simple. In the following lemma, the notation π(G) stands for the set of prime divisors of the order of G.

Lemma 2.2 Assume that G and S satisfy the hypothesis of Theorem 2.1. Then (1) |G| = |S|, (2) cd(G) = cd(S), (3) G is perfect, i.e., G = G0, (4) If N is a maximal normal subgroup of G, then G/N is a non-abelian simple group and cd(G/N) ⊆ cd(S). Furthermore, the ith-smallest nontrivial degree of G/N is greater than or equal that of S and π(G/N) ⊆ π(S). For simple groups of Lie type which is not the ﬁnite simple classical groups, we managed to prove the following result, which is the main part of the proof of Theorem 2.1 for these groups.

Proposition 2.3 Let S be a sporadic simple group, the Tits group, an alternating group of degree at least 7 or a finite simple exceptional group of Lie type and let T be any non-abelian simple group. If cd(T ) ⊆ cd(S), then T ∼= S. For each of the sporadic simple groups, the Tits group and the alternating groups of degree at least 5, and each possibility of the non-abelian simple group T, we compare several smallest nontrivial degrees of S and T using results of F. Lübeck [24] and R. Rasala [41] and also the classification of prime power character degrees of quasi-simple groups by G. Malle and A. E. Za- lesskii [27] to eliminate all but the simple groups which are isomorphic to the Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 6 given simple group S. For the exceptional simple groups of Lie type, apart from the previous results, we also made use of the explicit list of character degrees of these simple groups by F. Lübeck [25]. Let G be a finite group and let S be any non-abelian simple group which appears in Proposition 2.3 such that cd∗(G) = cd∗(S). By Lemma 2.2(4), cd(G/N) ⊆ cd(S) and G/N is non-abelian simple. Proposition 2.3 yields that G/N ∼= S and thus |G/N| = |S|. By Lemma 2.2(1), we obtain that |G| = |S| and so |N| = 1, which implies that G ∼= S. This gives a proof of Theorem 2.1 for the simple group S. For finite simple classical groups of Lie type, we can prove Proposition 2.3 for these groups by a similar method. However the proof is quite long and complicated. Instead, we have used a different approach. Assume now that S is a finite simple classical group in characteristic p and G is a finite group such that cd∗(G) = cd∗(S). With the same method as above, we can deduce that G/N is a finite simple group of Lie type in the same characteristic p. If N is trivial, then G is a finite simple group of Lie type in characteristic p, cd(G) = cd(S) and |G| = |S|. Using Artin’s Theorem mentioned earlier, we have {G, S} = {PSp2n(q), Ω2n+1(q)}, where n ≥ 3 and q is an odd prime power or {G, S} = {PSL4(2), PSL3(4)}. The latter case can be eliminated easily. For the former case, using the existence of the Weil characters of

PSp2n(q) with odd prime power q and the minimal characters of Ω2n+1(q) in [42, 43], one sees that these two groups have diﬀerent degree sets and thus G ∼= S in this case. Assume that N is nontrivial. As the Steinberg character StS of S of degree |S|p is the only irreducible character of S of nontrivial p-power degree and StG/N ∈ Irr(G/N) also has nontrivial p-power degree which is |G/N|p we deduce that |G/N|p = |G|p. If N is non-solvable, then N possesses a nontrivial irreducible character ϕ which is extendible to ϕ0 ∈ Irr(G) (see [8, Lemma 5]) and then by Gallagher’s Theorem [17, Corollary 6.17], we have ϕ0StG/N ∈ Irr(G) and hence ϕ0(1)|G/N|p = ϕ(1)|S|p ∈ cd(S), which is impossible as StS is the only irreducible character of S of p-defect zero (see [12, Theorem 4]). Recall that an irreducible character χ ∈ Irr(G) is of r-defect zero for some prime r if χ(1)r = |G|r. Therefore, N must be a nontrivial solvable group. In order to arrive at a contradiction, we need the following result.

Lemma 2.4 Let S be a finite simple group of Lie type and let G be a finite group. If cd(G) = cd(S) and |G| = |S|, then the Fitting subgroup F (G) of G is trivial. The proof of this lemma is an easy application of the existence of blocks of Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 7 defect zero of finite simple groups of Lie type [56] and Ito’s Theorem [17, Theorem 6.15]. Return to our problem, we see that F (G) is nontrivial as N is a nontrivial solvable subgroup of G and furthermore cd(G) = cd(S) and |G| = |S|. Now Lemma 2.4 will provide a contradiction. Problem 1.1 is now done for finite simple groups. The next natural groups to be considered are perhaps the direct product of copies of a non-abelian simple group.

Question 2.5 Let G be a ﬁnite group and H a direct product of copies of a non-abelian simple group such that cd∗(G) = cd∗(H). Can we conclude that G ∼= H?.

3 Quasi-simple Groups and Their Degree Multisets

We have seen in Section2 that every simple group is determined uniquely up to isomorphism by its degree multiset. What other groups have the same property? In [45], it was proved that all the symmetric groups are determined by their degree multisets, improving an old result of H. Nagao [30] that if G is a ﬁnite group whose character table agrees, up to a permutation of its rows and columns, with the character table of the symmetric group Sn, then ∼ G = Sn. This suggests that ﬁnite groups ‘close’ to simple might be determined by their degree multisets. We have proposed in [34]:

Conjecture 3.1 Let G be a finite group and H a finite quasi-simple group. If cd∗(G) = cd∗(H), then G ∼= H. In other words, every finite quasi-simple group is determined uniquely up to isomorphism by its degree multiset. Let us recall that a finite group H is said to be quasi-simple if H is perfect and H/Z(H) is non-abelian simple, in which case we also say that H is a perfect central cover or simply a cover of H/Z(H). Conjecture 3.1 indeed has been predicted earlier by the first author in [33], where he proved that every quasi-simple classical group H is uniquely determined up to isomorphism by its degree multiset, except possibly when H/Z(H) is isomorphic to PSL3(4) or PSU4(3). The main ideas can be sum- marized as follows. As H is perfect, it has a unique linear character. Therefore G has a unique linear character as well and hence G is perfect. It follows that, if M is a maximal normal subgroup of G then G/M is non-abelian simple. The first Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 8 step in the proof is quite similar to what we have done for simple groups; that is, to show that

G/M is isomorphic to S := H/Z(H).

This basically eliminates the involvement of all non-abelian simple groups other than S in the structure of G. Let Schur(S) denote the Schur cover (or the covering group) of S. We have

cd∗(G/M) ⊆ cd∗(G) = cd∗(H) ⊆ cd∗(Schur(S)), where the last containment comes from the fact that every cover of S is a quotient of the Schur cover of S. This condition together with some others if necessary can be used to force two non-abelian simple groups G/M and H/Z(H) to be isomorphic. On the way to the proof of Conjecture 3.1 for quasi-simple classical groups, we in fact prove the following:

Proposition 3.2 Let S be a simple classical group. Let G be a perfect group such that |S| | |G| | |Schur(S)| and cd(S) ⊆ cd(G) ⊆ cd(Schur(S)). If M is a maximal normal subgroup of G, then G/M ∼= S. The proof of this proposition depends heavily on the results on prime power representation degrees, due to G. Malle and A. E. Zalesskii [27], and relatively small character degrees of quasi-simple classical groups, due to P. H. Tiep and A. E. Zalesskii [42] and H. N. Nguyen [31]. The second step is to show that G ∼= H. Since |G| = |H| and G/M ∼= H/Z(H), we deduce that |M| = |Z(H)|. It follows that, if H is simple then M is trivial and we have immediately that G ∼= H. However, since we are working with quasi-simple groups, the problem becomes more difficult; especially for quasi-simple groups with complicated centers such as the covers PSL3(4) or PSU4(3) whose the Schur multipliers are very exceptional. We have done this in a series of lemmas. We reproduce the proofs of some of them. The first two lemma are straightforward by using the classification of finite simple groups.

Lemma 3.3 Let S be a non-abelian simple group. Let A be an abelian group such that |A| ≤ |Mult(S)|. Then |S| > |Aut(A)|.

Lemma 3.4 Let S be simple group of Lie type. Then no proper multiple of StS(1) is a degree of Schur(S). Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 9

For each nonnegative integer i, we let M (i) denote the ith derived subgroup of M.

Lemma 3.5 Let S be a non-abelian simple group. Let G be a perfect group ∼ and M C G such that G/M = S and |M| ≤ |Mult(S)|. Then, for every nonnegative integer i, the quotient G/M (i) is isomorphic to a quotient of Schur(S).

Proof We prove by induction that G/M (i) is isomorphic to a quotient of Schur(S) for every i. The induction base i = 0 is exactly the hypothesis. As- (i) ∼ suming that G/M = Schur(S)/Zi for some normal subgroup Zi of Schur(S), we will show that G/M (i+1) is also a quotient of Schur(S). As M (i)/M (i+1) is abelian and normal in G/M (i+1), we have

M (i) M (i) G ≤ C (i+1) ( ) . M (i+1) G/M M (i+1) E M (i+1)

(i) (i+1) (i+1) (i) (i+1) We ﬁrst consider the case CG/M (i+1) (M /M ) = G/M . Then M /M is central in G/M (i+1). As G is perfect, G/M (i+1) is a stem extension of (i) ∼ G/M = Schur(S)/Zi. As Schur(S)/Zi is a quasi-simple group whose quotient by the center is S, we deduce that G/M (i+1) is a quotient of the Schur (i+1) cover of Schur(S)/Zi. Therefore, G/M is a quotient of Schur(S), as wanted. (i) (i+1) The lemma is completely proved if we can show that CG/M (i+1) (M /M ) cannot be a proper normal subgroup of G/M (i+1). Assume so, then it follows by the induction hypothesis that

(i) (i+1) (i+1) C (i+1) (M /M ) G/M G/M ∼ G Schur(S) (i) (i+1) C (i) (i+1) = (i) = . M /M M /M M Zi Therefore,

(i) (i+1) C (i+1) (M /M ) Mult(S) M G/M (i) (i+1) ≤ = (i) M /M Zi M and hence (i) (i+1) (i+1) |CG/M (i+1) (M /M )| ≤ |M/M |. Thus G/M (i+1) ≥ |G/M| = |S|. (i) (i+1) CG/M (i+1) (M /M ) Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 10

Since the quotient group on the left side can be embedded in Aut(M (i)/M (i+1)) and M (i)/M (i+1) is abelian of order less than or equal to |M|, this last in- equality leads to a contradiction by Lemma 3.3. Since the Schur multipliers of the alternating groups and sporadic simple groups are ‘small’, the proof for these groups are easier and therefore, let us just focus on simple groups of Lie type.

Lemma 3.6 Let S be a simple group of Lie type. Let G be a perfect group and ∼ M C G such that G/M = S, |M| ≤ |Mult(S)|, and cd(G) ⊆ cd(Schur(S)). Then G is isomorphic to a quotient of Schur(S).

Proof By Lemma 3.5, we are done if M is solvable. So it remains to consider the case when M is nonsolvable. If M is nonsolvable, there is an 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 for some non-abelian simple group T . By [29, Lemma 4.2], T has a non-principal irreducible character ϕ that extends to Aut(T ). Now [8, Lemma 5] implies that ϕk extends to G/N. Therefore, by Gallagher’s lemma, ϕkχ ∈ Irr(G/N) for every χ ∈ Irr(G/M (i)). In particular,

ϕ(1)kχ(1) ∈ cd(G/N) ⊆ cd(G) ⊆ cd(Schur(S)).

Taking χ to be the Steinberg character of S. By Lemma 3.5, S is a quotient of G/M (i) and hence χ can be considered as a character of of G/M (i). We k now get a contradiction since ϕ(1) χ(1), which is larger than χ(1) = StS(1), can not be degree of Schur(S) by Lemma 3.4.

Lemma 3.7 Let S be a simple group of Lie type diﬀerent from S = PSL3(4) ∼ and PSU4(3). Let G be a perfect group and M C G such that G/M = S, |M| ≤ |Mult(S)|, and cd(G) ⊆ cd(Schur(S)). Then G is uniquely determined (up to isomorphism) by S and the order of G.

Remark 3.8 The same statement does not hold when S = PSL3(4) or PSU4(3). For instance, let S = PSL3(4). Then Mult(S) = Z4 × Z4 × Z3. Let Z1 and Z2 be subgroups of Mult(S) isomorphic respectively to Z4 and Z2 × Z2. The non-isomorphic groups Schur(S)/Z1 and Schur(S)/Z2 (see [10] where these groups are denoted by 121.S and 122.S) both satisﬁes the hypothesis of the lemma. Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 11

+ + Proof First we consider the case S = PΩ8 (2), Suz(8), or PΩ2n(q) with n ∼ even and q odd. Then Mult(S) = Z2 × Z2. By Lemma 3.6, S is isomorphic to a quotient of Schur(S) so that we can assume G ∼= Schur(S)/Z with Z ≤ Mult(S) (note that Z cannot be Schur(S)). If |M| = 1 or 4 then Z = Mult(S) or 1, respectively, and so we are done. Thus it remains to consider |M| = 2. We then have |Z| = 2 and hence Z is generated by an involution of Mult(S). However, as these three involutions are permuted by an outer automorphism of S of degree 3, the quotient groups of the form Schur(S)/hti for any involution t ∈ Mult(S) are isomorphic and we are done 2 again. Next, we assume that S = PSU6(2) or E6(2). Though the Schur multipliers of these groups are more complicated, these cases in fact can be argued similarly as above. If S is none of the groups already considered and also S 6= PSL3(4) and PSU4(3), then Mult(S) indeed is cyclic. Again, S is isomorphic to a quotient of Schur(S) and we can assume G ∼= Schur(S)/Z, where Z ≤ Mult(S). As G/M ∼= S, we then deduce that |Z| = |Mult(S)|/|M| = |Schur(S)|/|G|. Since the cyclic group Mult(S) has a unique subgroup of order |Schur(S)|/|G|, Z is uniquely determined by S and |G| and therefore the lemma follows. Following the method outlined above, we have shown in [34] that:

Theorem 3.9 Every finite quasi-simple group except possibly the Schur double covers of the alternating groups is uniquely determined up to isomorphism by its degree multiset. For now, we are unable to establish the first step for the Schur double covers of the alternating groups. Specifically, we do not know yet how to eliminate the case where H = Schur(An) and G/M is a simple group of Lie type in even characteristic. There is an ongoing work on this in [6] where the authors also study Prob- lem 1.1 for the Schur double covers of symmetric groups. The symmetric ˆ− group Sn has two isomorphism classes of Schur double covers, denoted by Sn ˆ+ ˆ+ ˆ− and Sn . It is well known that the group algebras CSn and CSn are canon- ˆ+ ˆ− ically isomorphic and therefore Sn and Sn are not uniquely determined by their degree multisets. Nevertheless, it is anticipated in [6] that

∗ ˆ+ ∗ ˆ− ∼ ˆ+ ˆ− if cd (G) = cd(Sn ) = cd (Sn ) then G = Sn or Sn . Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 12

The proof of this, as expected, depends heavily on the representation theory of the symmetric and alternating groups, their Schur double covers, and quasi- simple groups in general. One particularly needs the results on relatively small degrees and prime power degrees of the regular and spin representations of the alternating and symmetric groups. These results are due to various authors, including A. Balog, C. Bessenrodt, J. B. Olsson, and K. Ono [4], C. Bessenrodt and J. B. Olsson [7], and A. Kleshchev and P. H. Tiep [21, 22]. A group is said to be almost simple if it contains a non-abelian simple group and is contained within the automorphism group of that simple group. We end this section by a question.

Question 3.10 What are almost simple groups that are uniquely determined (up to isomorphism) by their degree multisets?

4 Character Degrees of Simple Groups and Huppert Conjecture

In Sections2 and3, we have seen that the degree multiset of a ﬁnite group encodes a lot of structural information of the group. In this section and the next one, we will drop the multiplicities of the character degrees and focus on the degrees only. The multiplicities will be considered in Section6. We will see that the degree set also provide some information on the group structure. In general, the character degree set of G does not completely determine the structure of G. As abelian groups have only the trivial character degree, it is easy to see that cd(G) = cd(G × A) for any abelian group A. There are many other examples without an abelian direct factor. For instance, the non- isomorphic groups D8 and Q8 not only have the same set of character degrees, but also share the same character table. The degree set also cannot be used to distinguish between solvable and nilpotent groups, as the groups Q8 and S3 show. It is unknown whether the degree set can distinguish solvable and non-solvable groups. Huppert conjectured in the late 1990s that the non-abelian simple groups are essentially determined by the set of their character degrees. More explicitly, he proposed in [15] the following.

Conjecture 4.1 (Huppert Conjecture) Let S be any non-abelian simple group and let G be a group such that cd(G) = cd(S). Then G ∼= S × A, where A is abelian. Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 13

As the character degrees of S ×A are the products of the character degrees of S and those of A, this result is the best possible. The hypothesis that S is a non-abelian simple group is critical. There cannot be a corresponding result for solvable groups. For example, if we consider the solvable group Q8, then cd(Q8) = cd(S3) but Q8 S3 × A for any abelian group A. Huppert in [15, 16] and his unpublished preprints, verified the conjecture on a case-by-case basis for many non-abelian simple groups, including the Suzuki groups, many of the sporadic simple groups, and a few of the simple groups of Lie type. Except for the Suzuki groups and the family of simple linear groups PSL2(q), Huppert proved the conjecture only for specific simple groups of Lie type of small, fixed rank. He indeed provided a pattern to approach the conjecture. (1) Show that G0 = G00. (2) Suppose that G0/M is a chief factor of G. Show that G0/M ∼= S. (3) Show that any linear character θ ∈ Irr(M) is stable under G0, which implies [M,G0] = M 0. (4) Show that M is trivial. ∼ 0 0 (5) Show that G = G × CG(G ). This pattern and variants thereof have been successfully used to make sig- nificant progress on the verification of the conjecture for a number of families of simple groups, notably the sporadic simple groups, alternating groups of small degree, and simple groups of Lie type of small rank. It is hoped that more general techniques can be developed to aid in the verification of the conjecture for simple groups of Lie type of higher rank and alternating groups of higher degree. T. P. Wakefield verified Huppert conjecture for most of the simple groups of Lie type of rank two as part of his dissertation research. These results appear in [57, 58, 59]. The conjecture was verified for G2(5) by S. H. Alavi and A. Daneshkah in [1] and established more generally for G2(q) by Wakefield and the second author in [54]. The conjecture for simple group F4(2) was verified in [55]. The remaining sporadic simple groups also have been shown to satisfy the conjecture in [2,3, 52]. In [44, 53], the result is established for the Steinberg triality and the Ree groups. For the alternating groups, it has been confirmed up to the degree 13 in [35]. We summarize all these results in the following theorem.

Theorem 4.2 Let G be a ﬁnite group and S a sporadic simple group, an alternating group of degree at most 13, a simple group of Lie type of rank at Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 14

∼ most 2 or F4(2). If cd(G) = cd(S), then G = S × A, where A is an abelian group. Let us now describe in more details some techniques as well as difficulties in accomplishing the steps in Huppert’s method. 1. The first step is to show that if S is a non-abelian simple group and G is a group such that cd(G) = cd(S), then G0 = G00. This step can be done by using the techniques in [15] and is described in details in [57]. Suppose that G00 < G0, and take K to be maximal subject to K being normal in G and G/K being a non-abelian solvable group. This means G0/K is the unique minimal normal subgroup of G/K. The structure of finite groups with this property has been described explicitly in [17]. In particular, it then can be deduced that G/K is either a p-group or a Frobenius group whose kernel is an elementary abelian p-group for some prime p. With this special structure of G/K, we can identify a degree of G that is not a degree of S to get a contradiction. 2. The second step is to show that if G0/M is a chief factor of G then it is isomorphic to the simple group S. By Step 1, the quotient G0/M is a non-abelian chief factor of G and therefore it is the direct product of k copies of a non-abelian simple group T :

G0/M ∼= T × T × · · · × T . | {z } k times To prove Step 2, we must show that k = 1 and T ∼= S. Huppert’s proofs of this step for some non-abelian simple groups mentioned above relies upon either very specific properties of the groups (for the Suzuki groups Suz(q) and the linear groups PSL2(q)) or the fact that the orders of the simple groups under consideration are divisible by very few primes. The approach by Wakefield and the authors for Step 2 in [35, 36], described as follows, has proved to be effective for more complicated simple groups, especially finite groups of Lie type of higher rank. If α is an irreducible character of S that extends to Aut(S), then, by tensor induction, it can be shown that

α × α × · · · × α is in Irr(G0/M) and extends to G/M. | {z } k times It in particular follows that α(1)k ∈ cd(G) and therefore

α(1)k ∈ cd(S). Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 15

If T is a simple group with “small” outer automorphism group such as alternating groups and sporadic groups, this will give many different characters α so that α(1)k is a degree in cd(S). Since it is unlikely that cd(S) will contain many degrees that are powers whose exponents are divisible by k, it should be possible to show that k = 1. The other possibility to consider is when T is a simple group of Lie type. We know that the Steinberg character St of T extends to Aut(T ). This yields St(1)k as a degree in cd(S). Recall that St(1) is a power of p, where the prime p is the defining characteristic for S. It is known that prime powers as degrees of quasi-simple groups are relatively rare [27]. Other than a known finite list, the only possibilities are the Steinberg characters of groups of Lie type. Unless S is one of the exceptions, this forces St(1) ∈ cd(T ) to be the degree of the Steinberg character of S. From this, we can obtain a bound on the characteristic and size of the underlying field of T . We then find a degree of T which divide degrees of S of relatively large degree and this will allow us to establish a contradiction if T S. 3. In Step 3, we have to prove a technical result that every linear character 0 of M is invariant in G . Let θ ∈ Irr(M) be such a character and let I = IG0 (θ) be the stabilizer of θ under the action of G0 on Irr(M). If θ is not invariant in G0, the inertia group I would be a proper subgroup of G0. This means I would be contained in a maximal subgroup, say U, of G0. Suppose that

I X θ = φi, where φi ∈ Irr(I). i

G0 0 0 0 We then have φi ∈ Irr(G ) and hence φi(1)|G : I| ∈ cd(G ) by Cliﬀord theory. It follows that

0 φi(1)|G : U||U : I| divides some degree of G. In particular, the index of U/M in G0/M (∼= S, by Step 2) divides some degree of G. We have seen that, in Step 3, knowledge of maximal subgroups of ﬁnite simple groups plays an important role. In particular, we need to consider the maximal subgroups of S that have indices dividing degrees in cd(S). The idea is to show that if U/M is a maximal subgroup of G0/M, then U is not the stabilizer of any character in Irr(M). Fortunately, it seems rare that S has maximal subgroups whose indices divide degrees in cd(S). 4. The result obtained in Step 3 implies in particular that [M,G0] = M 0 and |M : M 0| divides the order of Mult(G0/M) = Mult(S). We recall that Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 16

Mult(G) denote the Schur multiplier of G. In order to prove that M is trivial or equivalently G0 ∼= S, we have to analyze the diﬀerences between character degrees of various central extensions of the simple group S. In other words, the knowledge on the degrees of the irreducible projective representations of S is crucial in this step. This is expected to be complicated when the Schur multiplier of S is large. 5. The proof of Step 5 requires an understanding of the action of the automorphism group of a quasi-simple group on its irreducible representations. 0 0 Let C := CG(G ) be the centralizer of G in G. Then G/C is embedded into the automorphism group of G0 ∼= S. One sees that G/G0C will correspond to a subgroup of the outer automorphism group of S. The goal is to show that any outer automorphism of G0 will cause fusion among the irreducible characters in Irr(G0). This will produce a character degree in cd(G) that is not a degree in cd(S). In other words, we aim to show that

G = G0C.

Once this is proved, G will be a direct product of G0 and C. It follows that C ∼= G/G0 and hence C is abelian, as wanted. Step 5 is expected to be difficult when the outer automorphism group Out(S) of S is complicated. For instance, if S is a simple group of Lie type, the structure of Out(S) is d · f · g, where d is the group of diagonal automorphisms, f is the (cyclic) group of field automorphisms (generated by a Frobenius automorphism), and g is the group of graph automorphisms (coming from automorphisms of the Dynkin diagram). In order to understand the action of Out(S) on the set of irreducible characters of H, one has to study the action of these kinds of automorphisms individually. This topic might be of independent interest and we hope that Huppert conjecture will motivate more study on the behavior of irreducible representations of a simple group under the action of its outer automorphisms. Together with Wakefield, we have recently succeeded in applying these arguments to establish the conjecture for the simple linear and unitary groups in dimension 4. The result for the linear groups appears in [36] and its proof requires modifications in the five steps outlined above.

Theorem 4.3 ([36]) Let q ≥ 13 be a prime power and let G be a ﬁnite group such that cd(G) = cd(PSL4(q)). Then G is isomorphic to the direct product of PSL4(q) and an abelian group. Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 17

With further modiﬁcations of Huppert’s method, the second author has recently completed the veriﬁcation of Huppert conjecture for the remaining simple exceptional groups of Lie type in [51].

5 Extending Huppert Conjecture from Simple Groups to Quasi-simple Groups

Recent success on Problem 1.1 for quasi-simple groups suggests that Huppert conjecture might be extended from non-abelian simple groups to quasi-simple groups. In an ongoing work with Wakeﬁeld [37], we put forward the following, which is Conjecture 1.2 in the Introduction.

Conjecture 5.1 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. As mentioned in Section4, Huppert [15] outlined a pattern consisting of five steps to study his conjecture. This pattern and variants thereof have been successfully used in verifying the conjecture for several non-abelian simple groups. Drawing upon his method, the following pattern is proposed in [37] to approach Conjecture 5.1. (1) Show that G0 = G00. (2) Suppose that G0/M is a chief factor of G. Show that G0/M ∼= H/Z(H). (3) Show that G0 is isomorphic to a perfect central cover of H/Z(H). 0 0 0 (4) Show that G = G ◦CG(G ). It follows in particular that cd(G) = cd(G ). (5) Show that covers of H/Z(H) have distinct sets of character degrees. Steps 3 and 4 then imply that G0 ∼= H and hence G is isomorphic to 0 the central product of H and the abelian group CG(G ). It is worth to point out that Steps 1, 2, and 4 in pattern for quasi-simple groups are similar and respectively corresponding to Steps 1, 2, and 5 in Huppert’s method for non-abelian simple groups while Steps 3 and 5 are fundamentally different. With extra work, the proof of Huppert conjecture for a certain non-abelian simple group can be extended to obtain the proof of Conjecture 5.1 for the perfect central cover covers of that simple group, although it is not always obvious. In [37], by following the pattern outlined above, the authors have Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 18 established Conjecture 5.1 for all quasi-simple linear groups in dimensions 2 and 3.

6 Characterizing non-abelian simple groups by their multiplicity patterns

In this last section, we shift our focus on the multiplicities of character degrees of ﬁnite groups. Before we can present some results and open conjectures, we need some notation. For a ﬁnite group G, we write cd(G) = {d0, d1, ··· dt} with d0 = 1 < d1 < ··· < dt. For each positive integer d, the multiplicity of d in G, denoted by mG(d), is the number of irreducible characters of G having the same degree d, that is

mG(d) = |{χ ∈ Irr(G): χ(1) = d}|. The multiplicity pattern mp(G) of G is deﬁned to be the vector

(mG(d0), mG(d1), ··· , mG(dt)). Clearly, the first coordinate of mp(G) is the number of linear characters of G, which is |G : G0|, and for i ≥ 1, the (i + 1)th-coordinate of mp(G) is the multiplicity of the ith-smallest nontrivial character degree of G. Similarly, we can define mp1(G) to be the vector (mG(d1), mG(d2), ··· , mG(dt)). If the complex group algebra CG of G is given, then both mp(G) and mp1(G) are known. Also, if we know mp(G), then k(G), the number of conjugacy classes of G, can be computed by taking the sum of all entries of mp(G). It seems that the multiplicities of character degrees of a finite group G also have strong influence on the structure of the group. For example, it was proved by Moretóand Craven in [11, 29] that the order of a finite group G is bounded above by a function of the maximum multiplicity of character degrees of G. A conjugacy class analogue of this was studied by the first author in [32]. In [5], Berkovich and Kazarin proved that the nonsolvable groups in which only two nonlinear irreducible characters have equal degrees are exactly PSL2(5) and PSL2(7). This result has recently been generalised in [50], where it was showed that two simple groups above are also the only nonsolvable groups which have a unique nontrivial multiplicity of nontrivial character degrees. In [49], we proposed the following problem which is much stronger than Problem 1.1.

Problem 6.1 Given a ﬁnite group G, determine all ﬁnite groups (up to isomorphism) having the same multiplicity pattern mp(G) as that of G. Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 19

We can also ask the following question: What does mp(G) (or mp1(G)) know about G? Notice that we use the vector mp(G) rather than the set of multiplicities as there are many non-isomorphic groups having the same set of multiplicity of character degrees, for instance, all the groups D8,Q8, S3, A5, PSL2(7), J2,M22 and B have the same set of multiplicities which is {1, 2}. As both the quaternion group Q8 and the dihedral group D8 have the same character table, we deduce that mp(Q8) = mp(D8), which is equal to (4, 2). Hence, solvable groups are not uniquely determined by the multiplicity patterns. Using [10], we see that the multiplicities of the non-abelian simple groups in the Atlas are all distinct.

Conjecture 6.2 Let H be a non-abelian simple group. If G is a ﬁnite group such that mp(G) = mp(H), then G ∼= H. This conjecture, if true, would be a generalisation of Theorem 2.1. We do ∼ not know whether G = H or not if mp1(G) = mp1(H), with H a non- abelian simple group. Notice that mp1(S6 · 2) = mp1(S6). In supporting Conjecture 6.2, we have proved the following:

Theorem 6.3 Let G be a finite group and let H be a non-abelian simple group with at most seven distinct character degrees. If mp(G) = mp(H), then G ∼= H. The non-abelian simple groups with at most seven distinct character degrees have been classified by Malle and Moretó[26].

Lemma 6.4 ([26, Theorem C]) Let S be a non-abelian simple group. If |cd(S)| ≤ 7, then one of the following cases holds. ∼ f (1) |cd(S)| = 4 and S = PSL2(q), q = 2 ≥ 4. ∼ f (2) |cd(S)| = 5 and S = PSL2(q), q = p > 5 and p > 2. ∼ 2 2 2 2m+1 (3) |cd(S)| = 6 and S = B2(q ), q = 2 , m ≥ 1 or PSL3(4). ∼ (4) |cd(S)| = 7 and S = PSL3(3), A7, M11 or J1. In the next lemma, we obtain some consequences under the assumption that mp(G) = mp(H), where H is a ﬁnite perfect group and G is a ﬁnite group.

Lemma 6.5 Let G be a group and let H be a nontrivial perfect group. As- sume that mp(G) = mp(H). Then the following hold. (1) |cd(G)| = |cd(H)| and k(G) = k(H). Nguyen, Tong-Viet: Quasi-simple groups and their representation degrees 20

(2) G is perfect, i.e., G0 = G. (3) If M is a maximal normal subgroup of G, then G/M is a non-abelian simple group, k(G/M) ≤ k(H) and |cd(G/M)| ≤ |cd(H)|. Moreover, if d ∈ cd(G/M), then mG(d) ≥ mG/M (d). The proof of Theorem 6.3 is basically based on the previous two lemmas together with the explicit lists of the multiplicity patterns of those simple groups in Lemma 6.4. Indeed, if the ﬁnite groups G and H satisfy the hypotheses of Theorem 6.3, then by Lemma 6.5, G is perfect, k(G) = k(H), |cd(G)| = |cd(H)| and G/M is a non-abelian simple group with |cd(G/M)| ≤ |cd(G)|, where M is a maximal normal subgroup of G. It follows that G/M is one of the simple groups in Lemma 6.4 as |cd(G/M)| ≤ |cd(G)| ≤ 7. Now if |cd(G/M)| = |cd(G)|, then cd(G) = cd(G/M) and so by applying Theo- rem 4.2 and the fact that G is perfect, we obtain that G ∼= G/M. Hence, M = 1 and G is non-abelian simple with |cd(G)| = |cd(H)|. Using Lemma 6.4 again and the fact that G and H have the same number of conjugacy classes, we can deduce that G ∼= H as wanted. Assume that |cd(G/M)| < |cd(G)|. For each possibility of G/M, using Cliﬀord’s theory and the theory of character triple isomorphisms in [17], we can eliminate these cases by showing that condition mp(G) = mp(H) does not hold. For quasi-simple groups, we obtain the following.

Theorem 6.6 Let G be a ﬁnite group and let q ≥ 4 be a prime power. If ∼ mp(G) = mp(SL2(q)), then G = SL2(q). As with the complex group algebras, we conjecture that quasi-simple groups and the symmetric groups Sn can be uniquely determined up to isomorphism by their multiplicity patterns.

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