J. L. Alperin

Finite groups with quasi-dihedral and wreathed Sylow 2-subgroups.

The primary purpose of this paper is to give a complete classification of all finite simple groups with quasi-dihedral Sylow 2-subgroups. We shall prove that any such group must be isomorphic to one of the groups L3(q) with q= — \ (mod 4), U3(q) with q=\ (mod 4), or Mu. We shall also carry out a major portion of the corresponding classification of simple groups with Sylow 2-subgroups isomorphic to the wreath product of Z2n and Z2, n g: 2.

Sylow intersections and fusion

It is common in mathematics for a subject to have its local and global aspects; such is the case in group theory. For example, the structure and embedding of subgroups of a group G may be usefully thought of as part of the local structure of G while the normal subgroups, quotient groups and conjugacy classes are relevant to the global structure of G. Furthermore, the connections between local and global structure are very important. In the study of these relations, the methods of representation theory and transfer are very useful. The application of these techniques is often based upon results concerning the fusion of elements. (Recall that two elements of a subgroup H of a group G are said to be ficsed if they are conjugate in G but not in H.) Indeed, the formula for induced characters clearly illustrates this dependence. However, more pertinent to the present work, and also indicative of this connection with fusion, is the focal subgroup theorem [8]: if P is a Sylow p-subgroup of a group G then P n G’ is generated by all elements of the form a-lb, where a and b are elements of P conjzgate in G. Hence, this result, an application of transfer, shows that the fusion of elements of P determines P n G’ and thus P/P n G’ which is isomorphic with the largest Abelian p-quotient group of G. It is the purpose of this paper to demonstrate that the fusion of elements of a Sylow subgroup P is completely determined by the normalizers of the nonidentity subgroups of P. Therefore, P/P n G’, a global invariant of G, is completely described by the local structure of G. A weak form of our main result is as follows : if a and b are elements of a Sylow subgroup P of the group G and a and b are conjugate in G, then there exist elements a, ,..., a,,, of P and subgroups HI ,..., H,, of P such that a = a, , b = a, and ai and U~+~ are contained in Hi and cmjugate in N(H,), 1 < i < m 1. We shall strengthen

Weights for symmetric and general linear groups

Abstract An important feature of the theory of finite groups is the number of connections and analogies with the theory of Lie groups. The concept of a weight has long been useful in the modular representation theory of finite Lie groups in the defining characteristic of the group. The idea of a weight in the modular representation theory of an arbitrary finite group was recently introduced in Alperin (Proc. Sympos. Pure Math. 41 (1987, 369–379), where it was conjectured that the number of weights should equal the number of modular irreducible representations. Moreover, this equality should hold block by block. The conjecture has created great interest, since its truth would have important consequences—a synthesis of known results and solutions of outstanding problems. In this paper we prove the conjecture first for the modular representations of symmetric groups and second for modular representations in odd characteristic r for the finite general linear groups. In the latter case r may be assumed to be different from the defining characteristic p of the group, since the result is known when r is p. The well-known analogy between the representation theory of the symmetric and general linear groups holds here too.

LARGE ABELIAN SUBGROUPS OF p-GROUPS

For some time now, very little has been known about abelian subgroups of pgroups.We shall try and remedy this situation in a series of papers, of which this is the first. One impetus for doing this was created by several natural conjectures which arose in problems in other areas of group theory. In this paper we shall study p-groups with respect to the existence and nonexistence of abelian subgroups which in one sense or another can be considered as large subgroups. Later work will deal with several other topics. Our first result is in a negative direction; we shall demonstrate the falsity of the conjecture that every group of order pn has an abelian subgroup of order pn/2. Previously, we announced [1] that the best one could hope for was abelian subgroups of order at least p*48n, but for odd primes we can greatly improve this result by an entirely different method.

THE MAIN PROBLEM OF BLOCK THEORY

Publisher Summary This chapter discusses block theory, the primary aim of which is the discovery and verification of properties of the character table of a finite group. The method is that of congruences, of reduction modulo a prime p. It is often useful in character theory to rephrase statements in terms of the character table. In this case, the chapter discusses the submatrix consisting of the columns indexed by the conjugacy classes of p-singular elements and describes it readily in terms of the p-local subgroups. The chapter focuses on the extent to which the size of this submatrix of the character table is determined by the p-local subgroups.

Unipotent Conjugacy in General Linear Groups

ABSTRACT Let U(n,q) be the group of upper uni-triangular matrices in GL(n,q), the n-dimensional general linear group over the field of q elements. The number of U(n,q)-conjugacy classes in GL(n,q) is, as a function of q, for fixed n, a polynomial in q with integral coefficients.

Periodicity of Weyl modules for SL(2, q)

Weyl modules play a central role in the representation theory of the general and special linear groups. There has been great interest recently in the modular case of these groups over fields of prime characteristic p. In particular, Glover [l] has proved some remarkable and comprehensive results about these modules for SL(2,p). We shall extend one of these theorems to the case of SL(2, q) where q =pe. First, let us fix some notation. We set S = ,X(2, k), k a field of q elements and R = k[x, y] the polynomial algebra in two variables considered as a module for kS, the group algebra, in the usual way. Let V, be the kS-module of dimension n consisting of the homogenous polynomials of degree n 1. Hence, V, is the trivial kS-module and the V, are the duals of the Weyl modules for S. Let V, = IV,, + P, be a direct decomposition where P, is a projective kS-module and W,, has no non-zero projective direct summand. Our main result is as follows:

The Green correspondence and normal subgroups

In a recent paper, Harris and Knorr [6] have discovered a connection between the Brauer correspondence and the covering relation between blocks of a finite group and blocks of its normal subgroups. Green’s module point of view of blocks [4] suggests that this result has a moduletheoretic proof and that there is a more general result on indecomposable modules. We shall show that this is the case by giving a short moduletheoretic proof of the Harris-Kniirr theorem and then use the ideas of the proof to formulate and establish a result relating the Green correspondence [IS] with normal subgroups. Let us establish some notation. Let G be a finite group and H a normal subgroup. Let k be a field of prime characteristic p and b a block of H so b is an indecomposable algebra summand of kH. We shall regard b as a k[H x HI-module in the usual way. Let D be a defect group of b, K = N,(D) and b, equal the block of K with bf’ = b, that is, the Brauer correspondent of b. Finally, set L = N,(D) so K is a normal subgroup of L.

Transfer and fusion in finite groups

There exist a number of well-known theorems which give conditions under which the structure of the normalizer of a particular p-subgroup of a finite group G determines certain “global” properties of G, such as, the largest abelian p-factor group of G or the conjugacy of p-elements in G. For example, we may mention Burnside’s theorems on the conjugacy of elements in the center of a Sylow p-subgroup and on the existence of normal p-complements in groups with abelian Sylow p-subgroups. The theorem of Griin concerning p-normality and the Hall-Wielandt theorem are also of this nature. In this paper we shall establish a number of general results of this type. To state these, we must first introduce several concepts. Throughout the paper G will denote a fixed finite group, p will be a fixed prime divisor of the order of G, and Z will designate the set of all nonidentity p-subgroups of G.

Up and down fusion

In an earlier work [l] we have demonstrated the local nature of fusion of elements in a Sylow p-subgroup P of a group G. We showed that if x and y are elements of P which are conjugate in G then there is a sequence of elements x1 ,... , e~‘n of P such that x = x1 , y = x, and each pair xi, xifl is contained in a subgroup Hi of P and are conjugate in N(H,). The elements xi, xifl are said to be “locally conjugate” and so every conjugation is the result of a sequence of local conjugations. Previously, we gave great details about the nature of the subgroups Hi that could be used and of the elements of N(H,) which performed the actual conjugation of xi to X~+~ . However, in this sequel, it is the actual sequence xi ,..., x,~ that we shall see can be chosen in a very special way. This result is of use in proving that certain elements of P are not conjugate in G. We shall give an example of this at the end of this paper but the main applications will come in later work. Let us now state the main theorem in its simplest form. After proving this version of it we shall examine our proof and be able to state and digest the final form of our main result.