We outline the main steps in the classification of finite irreducible subgroups of G L(p, D), where pis a prime and D is a division ring of characteristic zero

Shirvani, M.

English

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Resenhas IME-USP 1996, Vol. 2, No.3, 313 - 317. On Finite Irreducible Subgroups of GL(p,D) 1 M. Shirvani Abstract: We outline the main steps in the classifica-tion of finite irreducible subgroups of G L(p, D), where pis a prime and D is a division ring of characteristic zero. Key words: Finite simple group, division ring, centre-by-finite group, irreducible representation. 1. INTRODUCTION, HISTORY, AND QUESTIONS What can one say about the structure of a finite group, given that it has a faithful irreducible representation of a given degree? In fact, there is more than one way of interpreting this question. The most comprehensive is via the classification of all finite subgroups of G L( n, K), for suitable fields K (usually the complex numbers) for successive values of n. The most up-to-date work seems to be that of Feit [4], where all finite groups having a finite irreducible quasi-primitive complex representation of degree 8 :S n :S 10 are classified. Since the earlier work of Blichfeldt, Brauer, Lindsey, and Wales (see Feit's article for references) had already covered the cases n :S 7, the classification is now complete up to n = 10. The correspondtng information , however , is much more limited for fields that are not algebraically closed . A second approach involves the study of groups having a faithful irreducible complex representation whose degree is small compared to some prime divisor of the order of the group. The best result to date is that of Zhang Jiping [14], who classified such groups G if the degree of the representation is :S p - 1, where p is a prime dividing the order of the group (earlier work had been done by Brauer, Than , Ferguson, and Feit). The proof uses the classification of finite simple groups, as well as the well-known fact that the Sylow p-subgroups of G are abelian (for the particular prime p), and this fact is quite crucial in the proof. The result can be paraphrased by saying that G has a normal subgroup H such that CG(H) ~ H, and H is a " known" group. The above results make essential use of the fact that the underlying field is commutative and algebraically closed, although it also becomes apparent that the groups in the families do not resemble each other as one moves from dimension n to n + 1. Now when dealing with small fields (such as the rationals), one has to worry about the Schur index of the representation as well as the matrix degree. Indeed, a simple component Dnxn of the rational group algebra of G corresponds to a family of (conjugate) complex representations of degree nm, where m is 1 Research partially supported by NSERC, Canada. AMS Classification: 20C15, 20H25. 313 314 M. Shirvani the corresponding Schur index. It turns out that, in this generality, the decisive factor influencing the structure of the group is the complexity of the degree n. In addition , one can use essentially the same techniques to study centre-by-finite groups having representations of particular degrees. This, of course, is the same as studying the projective representations of the central quotient group . Finite subgroups ofGL(l, D) = D* were classified by Amitsur [1], so one may turn to finite subgroups of G L(p, D), where D is a division ring of characteristic zero, and p IS a pnme. Question 1. For what finite groups G is there a decomposition of the rational group algebra QG = ... + DPxp + ... ? That is, what finite groups have at least one rational reduced degree equal to a prime? Much is known about the structure of the groups if all the reduced degrees are powers of the same prime (see, for example, the work of Huppert and Gow in [8] and [9]), but little is known about the question which is posed here, except when p = 2, where the classificatio.n was done by Banieqbal [2], based on earlier work of Hartley and Shahabi [6] (which in turn depended on Gorenstein and Harada's classification of groups of sectional 2-rank at most 4). We can make Question 1 more specific, as follows : Question 2. If k = Z(DPXP) is the centre of DPxp, and G is a (centre-by-finite subgroup of (DPXP) such that k[G] = DPxp, then what can be said about the structure of G and/or D? The project discussed in this paper aims, as a first step, at the classification ofthe finite groups G. In this case, the division rings D are well-known to belong to the Schur subgroup of the Brauer group of k, which was first determined by Yamada [13] for local and global fields k . If k is a local or a global field, then Lorenz and Opolka [10] show that every class of the Brauer group of k contains an algebra of the form k[G] = Dnxn , where G is centre-by-finite. Note that the n here is not necessarily a prime. Not much is known about the structure of such G, even when n = 1, although one at least knows the structure of G/(I(G) (where (l(G) denotes the centre of G) (see [11], where the finite subgroups of PGL(l, D) are classified) . For the rest of the paper, G is a finite subgroup of a central simple k-algebra DPxp = k[G], (i.e., G contains a k-basis for the algebra) . If N is any normal subgroup of G, then G acts transitively (by conjugation) on the simple components of k[N] ( [12], 1.1.6). 2. CLIFFORD THEORY AND MINIMALITY Suppose G has a normal subgroup N such that S = k[N] is not a simple rin~ On Finite Irreducible Subgroups of GL(p,D) 315 Then S is a direct sum of p copies of D, permuted transitively by G, and so Gis a subgroup of a permutational wreath product H wr Go, where H is a subgroup of D*, and Go is a primitive permutation group of degree p (so Go acts on the p copies of H through its action on the index set {I, ... ,p } ) . The structure of the Amitsur groups H is well-known (Amitsur [1], or see [12], 2.1.4 and 2.1.5). As for Go, we know that either Go is soluble or it is 2-transitive. In the first case, Galois already knew that Fp ::; Go ::;FpF; (e.g. [7], 11 .3.6 ), while the O'Nan-Scott Theorem together with the classification of finite simple groups implies that Go is one of the following (cf. [3], Theorem 5.3): • Alt(p) , Sym(p) , PSL(d, q) or PGL(d, q) with p = qqd~/. • PSL(2, 11) , PGL(2, 11), or Mll when p = 11. • M 23 when p = 23. d It is . not known whether there exist infinitely many pairs (q, d) for which qq~/ is a prIme. It may be useful to mention the idea of minimality at this point, as it is useful in setting up inductive arguments. Suppose that G has a proper normal subgroup N such that k[N] = k[G] = R, say. Then it can be shown that there exists a subgroup dR(N) of R such that N ::; G ::; dR(N), N is normal in dR(N), and if p(k) denotes the group of roots of unity in k*, t hen dR(N)/ p(k)N is a subgroup of the outer automorphism group OuteN). Therefore, once one knows the structure of Nand k[N] = DPxp, the structure of G may be effectively determined too. We say that G has a minimal embedding into k[G] = DPxp if k[N] =I k[G] for every proper normal subgroup N of G . We maynow suppose, for the rest of the paper, that for every proper normal subgroup N of G, the ring k[N] is simple and distinct from k[G]. Of course, this means that k[N] has the form t... or t...Pxp for some division ring t.... 3. COMPONENTS OF G Recall that a component of a finite group is a subnormal subgroup X =I 1 such that X = XI and X/(l(X) is a simple group (so non-abelian). In our situation, it can be shown that every component of G is actually normal in G, and so either X = SL(2, 5) generates a division ring, or k[X] = LlPXP for some Ll . It is then trivial to see that G can have at most one component of each type (otherwise the normal subgroup E( G) generated by all the components spans a matrix ring of degree divisible by p2). The next easiest case is when the Fitting subgroup of G is strictly greater than the centre of E(G). This is because (if q is a prime dividing the order of F = Fitt(G)) 316 M. Shirvani the Sylow q-subgroup Q of F can only generate simple rings of dimension a power of q. Therefore, Q is cyclic if q #- 2 or p, Q is cyclic or quaternion if q = 2, and Q is extra-special of order p3 if q = p (supposing that p is an odd prime) . These are of course immediate consequences of P. Hall's classification of p-groups all of whose characteristic abelian subgroup~ are cyclic ([7], III.13.10). It is then relativly easy (though very tedious) to classify G itself. This, in particular, gives the structure of all soluble G. Finally, the c~e where Fitt(G) = (1(E(G)) has to be handled. This requires detailed information about the automorphism groups of covering groups of finite simple groups and their representations. The following examples may provide an indication of the sort of argument that is required. 4. EXAMPLES Example 1. Let p = 3, and suppose G has a component X = 2.h, the covering group of the second Janko group . What can we say about G? It is known that the rational group ring of X has a simple component Q[X) = ~ 3X3 , where ~ is the quaternion algebra C;~:Ts~) (see, e.g., Feit [5)) . It follows from the structure of ~ that every cyclotomic field Q(en), where en is a primitive n-th root of unity with n 2:: 3, splits ~. This immediately implies that Fitt( G) = (1 (X) has order 2, so Ga(X) = (1(X). Therefore, X ::; G::; Aut(X). Since Out(X) has order 2, it only remains to check whether G = Aut(X) is a possibility. It is not , so G = X. Example 2. Suppose p = 23, and G has a component X = M24 (recall that the Schur multiplier of M24 is trivial) . There is a ring Q[X) = Q23X23 . Since Out(X) = 1, it is immediate that G = M24 X H, where H is an Amitsur group. Example 3. Suppose G is a Quasi-Amitsur group , i.e., G is not an Amitsur group, but every proper characteristic subgroup of G can be embedded into a division ring. Such groups may, in a sense, be regarded as generalizations of p-groups all of whose characteristic abelian subgroups are cyclic. These groups have very special properties, and can be classified. Many (though not all) have faithful representations of prime degree over division rings of characteristic zero. Acknowledgements. I wish to extend my sincere thanks to Cesar Polcino-Milies and Maria Lucia Sobral-Singer, whose huge organizational efforts did so much to make the conference a success, and to IME for their hospitality, and last but not least, to Jairo Gonc;alves, for being a gracious host, a great collaborator, and a good friend. On Finite Irreducible Subgroups of GL(p,D) 317 References [1] S. A. Amitsur , Finite subgroups of division rings, Trans. Amer. Math . Soc. 80 (1955), 361-386. [2] B. Banieqbal, Classification of finite subgroups of 2 x 2 matrices over a division algebra of characteristic zero, J. Algebra 119 (1988), 449-512. [3] P. J. Cameron, Finite permutation groups and finite simple groups, Bull. London math. Soc. 13 (1981) 1-22. [4] W. Feit, On finite linear groups in dimension at most 10, in Proceedings of the Conference on Finite Groups, Park City, Utah, 1975, eds. W. R. Scott and F. Gross, 397-407 (Academic Press, 1976) . [5] W. Feit, The computation of some Schur indices, Israel J. Math. 46 (1983), 274-300. [6] B. Hartley and M. A. Shahabi-Shojaei, Finite quasi-simple subgroups of 2 x 2 matrices over a division ring , Math. Proc. Camb. Phil. Soc. 97 (1985), 415-420. [7] B. Huppert , Endliche Gruppen (Springer, 1967). [8] B: Huppert, Research in representation theory at Mainz, in Representation Theory of Finite Groups and Finite-Dimensional Algebras, eds. G . O. Michler and C. M. Ringel 2-36 (Birkhauser, 1991). [9] B. Huppert and R. Gow, Degree problems of representation theory over ar-bitrary fields of characteristic 0, I and II, J. reine angew. Math. 381 (1987), 136-147, 389 (1988), 122-132. [10] F. Lorenz and H. Opolka, Einfache Algebren und projektiv Darstellungen uber Zahlenkorpern, Math. Zeit. 162 (1978), 175-182. [11] M. Shirvani, The finite inner automorphism groups of division rings, Math. Proc. Camb. Phil. Soc. 118 (1995), 207-213. [12] M. Shirvani and B. A. F. Wehrfritz Skew Linear Groups (Cambridge Univer-sity Press, 1987). [13] T. Yamada The Schur Subgroup of the Brauer Group (Springer, 1974). 318 M. Shirvani [14] J . Zhang, Complex linear groups of degree at most p -1, in Classical Groups and Related Topics, eds. A. J. Hahn, D. G. James, and Z.-X. Wan, 243-254 (Contemporary Mathematics, Volume 82). M. Shirvani Department of Mathematical Sciences University of Alberta Edmonton, AB, Canada T6G 2G1 mazi@schur.math.ualberta.ca Canada 

On Finite Irreducible Subgroups of GL(p,D)

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On Finite Irreducible Subgroups of GL(p,D)

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