I. Martin Isaacs

On groups of central type

A finite group is said to be of central type if it possesses an irreducible complex character which takes the value zero on all noncentral elements. (Equivalently, the degree of this character is the square root of the index of the center.) In 1964, Iwahori and Matsumoto [15] conjectured that a group of central type must be solvable. The paper of Liebler and Yellen [16] aims to prove this, but there is a gap in their proof (as explained below). Nevertheless, they do correctly obtain substantial information about a minimal counterexample to the conjecture. The aim of this paper is to fill the gap in the Liebler-Yellen proof (using as they do, the classification of finite simple groups) and to provide further information about which solvable groups can have central type. We show, for instance, that every normal subgroup of the central factor group of a (solvable) group of central type has the property that its system normalizers have square index. It is interesting to contrast this restrictive condition with the result of Gagola that every solvable group is embeddable in the central factor group of a group of central type. (See [6, Theorem 1.2]). The arguments in [16] show that a minimal counterexample G to the Iwahori -Matsumoto conjecture must have the following structure: If K/Z is any minimal normal subgroup of G/Z (where Z = Z ( G ) , the center), then K is abelian, K/Z is a 2-group, S = CG(K ) is the maximal solvable normal subgroup of G and any minimal normal subgroup HIS of G/S is the direct product of a number l of copies of GL(3, 2). Indeed, the Liebler-Yellen methods can be used to show that the Sylow 2-subgroup of K, considered as an (H/S)-module is a product o f / copies of a GL(3,2)-module of dimension 4, this module having a unique minimal submodule, which is trivial, and an irreducible factor module of dimension 3. The minimal submodules are amalgamated in the product.

Supercharacters, symmetric functions in noncommuting variables, and related Hopf algebras

We identify two seemingly disparate structures: supercharacters, a useful way of doing Fourier analysis on the group of unipotent uppertriangular matrices with coefficients in a finite field, and the ring of symmetric functions in noncommuting variables. Each is a Hopf algebra and the two are isomorphic as such. This allows developments in each to be transferred. The identification suggests a rich class of examples for the emerging field of combinatorial Hopf algebras.

Character stabilizer limits relative to a normal nilpotent subgroup

In his recent paper [ 11, Dade established the existence of some (rather unexpected) invariants of certain irreducible characters of finite groups. The purpose of the present paper is to state and to provide a relatively easy proof for a somewhat weakened and less general form of Dade’s result. In fact, Dade, in his introduction to [ 11, explicitly encouraged me to publish this simplified approach. To explain and motivate our theorem, we need to discuss “stabilizer limits.” Let x E Irr(G). (We work only with characters over the complex numbers. Dade, on the other hand, considers other characteristic zero fields too.) If Ma G and 8 is an irreducible constituent of xM, let T= Z&O), the inertia group. Then there is a unique character VE Irr(TI 0) such that qG =x. We call q the Clzfford correspondent of x with respect to 0. (Note that by taking M = 1, we see that x is one of its own Clifford correspondents.) We can repeat this process and consider Clifford correspondents (in T) for the Clifford correspondent q E Irr( T) of x. A character

Character theory of finite groups

arising via any number of such iterations, we call a compound ClifSord correspondent (CCC) of x and we denote the set of these objects by CCC(x). Note that if Ic/ ECCC(X), then Ic/“=x. Consider “minimal” element of CCC(x). These are the compound Clifford correspondents

Finite Group Theory

of x which themselves have no proper Clifford correspondents. (In other words, they are quasiprimitive.) Following T. Berger and Dade, we call these characters the stabilizer limits of x. What do the different stabilizer limits for a given x E Irr(G) have in common? In general, the answer is “not much”.