I. M. Isaacs

New refinements of the McKay conjecture for arbitrary finite groups

Let G be an arbitrary finite group and fix a prime number p. The McKay conjecture asserts that G and the normalizer in G of a Sylow p-subgroup have equal numbers of irreducible characters with degrees not divisible by p. The Alperin-McKay conjecture is version of this as applied to individual Brauer p-blocks of G. We offer evidence that perhaps much stronger forms of both of these conjectures are true.

Character correspondences and irreducible induction and restriction

Let S and G be linitc groups such that S acts on G and (IS\, ICI) = 1 and let C denote the fixed-point subgroup, C = C,(S). It is well known in this situation that there exists a natural bijection from the set Irr,(G) of S-invariant irreducible characters of G onto Irr( C), the full set of irreducible characters of C. When S is solvable, this bijection was constructed by G. Glauberman [2] and in the remaining case, where G is solvable of odd order, a totally different construction [3] gives the desired character correspondence. It was proved by T. R. Wolf [g] that when both constructions apply, when S is solvable and IG( is odd, the two constructions yield exactly the same map. If 31 E Irr,(G), we write x* E Irr(C) to denote the uniquely determined corresponding character. Similarly, if HE G is S-invariant, and 0 E Irr,(H). we write f3* E Irr(Hn C) to denote the image of 8 under the character bijection induced by the action of S on H. Our main result is the following.

Lie algebras with self-centralizing ad-nilpotent elements☆

Let L be a finite dimensional Lie algebra over an arbitrary field F. For x EL, the centraZizer C(x) is the subalgebra {y EL ( [yx] = 0). We say that x is selfcentralizing if dim C(x) = 1, ( i.e. C(x) = Fx). If L contains an ad-nilpotent self-centralizing element, then the structure of L is severely constrained. In Section 2 we shall see that if char(F) = 0, then dim L 0, then examples exist with dimension exceeding 3. It turns out for p > 2 that these algebras are necessarily simple and have p-power dimension. (The situation when p = 2 is more complicated and we do not discuss it.) Let char(F) = p > 2. We say that a polynomial f E F[Xj is a p-polynomial if the only powers of X having nonzero coefficients in f are of the form XP” for i > 0. Given a positive integer n and a p-polynomial f, we construct in Section 4 a specific Lie algebra La(f) of dimension p %. These algebras contain self-centralizing ad-nilpotent elements and for perfect fields L, we show in Section 7 that the L,(f) are the only algebras of dimension >3 which do. The definition of L,(f) is motivated by the special case f = 0 where the algebra can be identified with the Zassenhaus algebra of dimension pn (see Ree [S]). In Section 3, we study the structure of Zassenhaus algebras from the point of view of self-centralizing ad-nilpotent elements. A consequence of this investigation is an easy proof that when F is algebraically closed, the Zassenhaus algebras and o((2, F) are the only finite dimensional simple algebras which have subalgebras of codimension 1. (This was asserted by Amayo [2] without any assumption on F. \Ve give a counterexample in Section 6 which shows that the result is not true over arbitrary fields.) Another generalization of the Zassenhaus algebras was given by Albert and Frank [l]. In Section 5 we show that these “Alberttzassenhaus” algebras over perfect fields also contain self-centralizing ad-nilpotent elements and hence are

Partial characters of π-separable groups

Our concern in this expository paper is the character theory of a finite group G as seen from the perspective of a set π of prime numbers. There are no new results here and few new ideas. Our purpose is to present in as accessible a manner as possible, the proofs of some theorems in the character theory of π-separable groups, and to explain the significance of these results.

A note on groups of p-length 1∗

Abstract The finite group G is said to have p -length 1 if there exist normal subgroups H ⊆ K ⊆ G such that H and G K are p′ -groups and K H is a p -group. In this note, we give some character-theoretic characterizations of groups with p -length 1. These concern the restrictions of characters to a Sylow p -normalizer. We also prove a sufficient (but hardly necessary) condition for a p -solvable group to have p -length 1. This condition also involves the Sylow p -normalizer.

Lifting Brauer characters of p-solvable groups, II

Let IBr(G) denote the set of irreducible Brauer characters of the finite group G with respect to a fixed prime p and a fixed lifting of the p-modular roots of unity to the complex numbers. If 4 is a (complex valued) class function on G, we let I/J* denote the restriction of # to the p-regular elements of G. Thus if # is an (ordinary) character of G, then

Groups with just one character degree divisible by a given prime

* is a linear combination of IBr(G) with nonnegative integer coefficients. If G is p-solvable and v E IBr(G), then the Fong-Swan Theorem asserts that y = x* for some character x of G. (Necessarily, x E Irr(G), i.e. x is irreducible.) Our concern here is to choose the characters ,y in a consistent manner which respects restriction to normal subgroups. We shall construct for eachp-solvable group G, a uniquely defined subset oY(G) C Irr(G) such that:

Lie algebras with few centralizer dimensions

The Ito-Michler theorem asserts that if no irreducible character of a finite group G has degree divisible by some given prime p, then a Sylow p-subgroup of G is both normal and abelian. In this paper we relax the hypothesis, and we assume that there is at exactly one multiple of p that occurs as the degree of an irreducible character of G. We show that in this situation, a Sylow p-subgroup of G is almost normal in G, and it is almost abelian.

Fixed point spaces, primitive character degrees and conjugacy class sizes

Abstract It is known that a finite group with just two different sizes of conjugacy classes must be nilpotent and it has recently been shown that its nilpotence class is at most 3. In this paper we study the analogs of these results for Lie algebras and some related questions.

Counting objects which behave like irreducible Brauer characters of finite groups

Let G be a finite group that acts on a nonzero finite dimensional vector space V over an arbitrary field. Assume that V is completely reducible as a G-module, and that G fixes no nonzero vector of V. We show that some element g ∈ G has a small fixed-point space in V. Specifically, we prove that we can choose g so that dim C V (g) < (1/p)dim V, where p is the smallest prime divisor of |G|.