Alexandre Turull

Character Theory and Length Problems

We discuss some length questions in the theory of finite solvable groups for which character theory has proven useful. We discuss some open problems, some techniques of proof for various character and module theoretic results arising in this context, as well as the applications of these results to the length problems themselves. As this is an expository paper, most of our results appeared elsewhere. An exception is a linear bound on the Fitting height of groups accepting a fixed point free abelian group of automorphisms of square-free odd exponent, which, although it follows easily from known results, does not seem to have been noticed before.

The Schur index of projective characters of symmetric and alternating groups

The present paper answers a natural question that arises when one sets side by side two major contributions of Issai Schur to the theory of group representations. In a monumental paper [13], Schur described the projective characters of the symmetric group Sn and the alternating group An) for all n. He calculated the representation groups of Sn and An) for all n, and calculated the character table of these representation groups, thereby describing the projective characters of Sn and An. In another seminal paper [12], Schur defined and studied what is now called the Schur index. If 4f is an irreducible character of some finite group G and K is a field of characteristic zero, the Schur index mK(G) of qf with respect to K is the least positive multiplicity of q/ in a character afforded by a KG-module. The question is: What are the Schur indices of the characters of the representation groups of Sn and An? We give here a rule that describes the Schur index of every irreducible character of the representation groups of Sn and An) for all n. A substantial amount of research has been devoted to the Schur index. Efforts have been directed both to finding its general properties and to calculating it for important classes of examples. For example, Benard [1] showed that the Schur index of every character of the Weyl groups of type E6, E7 and E8 is one. In [2], Feit gives short proofs for the calculation of the Schur index for many specific groups, including Benards Weyl groups of type E6, E7 and E8, and the representation groups of the sporadic simple groups. Other authors have studied the Schur index for groups of Lie type; see for example [4] or [10]. Various methods are available to give upper bounds for the Schur indices, but it is usually harder to find their value when they are larger than one. Hence, for the groups of Lie type, one can show that their Schur indices are at most two in many cases and at most one in some cases, but the question of exactly for what characters it is one and for which it is two has not been consistently answered.

Some invariants for equivalent G-algebras

Abstract In an earlier paper (Clifford theory with Schur indices, J. Algebra 170 (1994) 661–677), the author introduced a generalization of the Brauer-Wall group. It is defined for any given finite group G and any field F of characteristic 0. Each element of this generalized Brauer-Wall group is an equivalence class of central simple G-algebras. He showed that given a finite group H with a normal subgroup N such that H N ≅ G , and an irreducible character χ of H, there corresponds naturally an element of this generalized Brauer-Wall group, and that this element alone controls the Clifford theory (including Schur indices) of χ with respect to N. The present paper studies some invariants for G-algebras which are preserved under equivalence of G-algebras in the above sense. These invariants are the basis of a characterization of each equivalence class of central simple G-algebras in some cases, as is described in a forthcoming paper of the author. The present paper also includes a brief comparison of this generalization of the Brauer-Wall group with others that have appeared in the literature.

P-rational characters and self-normalizing sylow p-subgroups

Let G be a finite group, p a prime, and P a Sylow p-subgroup of G. Several recent refinements of the McKay conjecture suggest that there should exist a bijection between the irreducible characters of p′-degree of G and the irreducible characters of p′-degree of NG(P ), which preserves field of values of correspondent characters (over the p-adics). This strengthening of the McKay conjecture has several consequences. In this paper we prove one of these consequences: If p > 2, then G has no non-trivial p′-degree p-rational irreducible characters if and only if NG(P ) = P .

Reduction theorems for Clifford classes

Abstract Clifford Theory concerns the representations over a field F of a finite group G and one of the normal subgroups N of G. It is often viewed as a series of reduction theorems. Clifford classes are certain equivalence classes of G/N-algebras over F. They have been used to describe the Schur indices of the irreducible characters of certain families of classical groups. In the present paper, we show how Clifford classes can also be used to explain and to reflect the reduction theorems of classical Clifford theory. We show that certain products of characters correspond to certain products of Clifford classes. We show that induction, restriction, inflation, and extension of the base field can all be defined naturally for Clifford classes. We prove that the properties of these elementary operations on Clifford classes imply all of the standard Clifford-theoretic reductions in the case when the field is ℂ. They also provide important information in the case when the field F is an arbitrary field of characteristic zero.

The number of Hall -subgroups of a -separable group

We observe a simple formula to compute the number νπ(G) of Hall π-subgroups of a π-separable finite group G in terms of only the action of a fixed Hall π-subgroup of G on a set of normal π′-sections of G. As a consequence, we obtain that νπ(K) divides νπ(G) whenever K is a subgroup of a finite π-separable group G. This generalizes a recent result of Navarro. In addition, our method gives an alternative proof of Navarro’s result.

Schur indices of perfect groups

It has been noticed by many authors that the Schur indices of the irreducible characters of many quasi-simple finite groups are at most 2. A conjecture has emerged that the Schur indices of all irreducible characters of all quasi-simple finite groups are at most 2. We prove that this conjecture cannot be extended to the set of all finite perfect groups. Indeed, we prove that, given any positive integer n, there exist irreducible characters of finite perfect groups of chief length 2 which have Schur index n.

Principally separated non-separated solvable groups

Abstract If π is a set of primes, a finite group G is called block π-separated if for every two distinct irreducible complex characters α, β ∈ Irr(G) there is a prime p ∈ π such that α and β are in different p-blocks. The group G is called principally π-separated if the above holds whenever β = 1 G . Bessenrodt and Zhang conjectured that if G is a solvable principally π-separated group then G is π-separated. We construct a family of counter-examples to this conjecture.

Cyclic by prime fixed point free action

Let the finite group A be acting on a (solvable) group G and suppose that no non-trivial element of G is fixed under the action of all the elements of A. Assume furthermore that (|A|, |G|) = 1. A long standing conjecture is that then the Fitting height of G is bounded by the length of the longest chain of subgroups of A. Even though this conjecture is known to hold for large classes of groups A, it is still unknown for some relatively uncomplicated groups. In the present paper we prove the conjecture for all finite groups A that have a normal cyclic subgroup of square free order and prime index. Since many of these groups have natural modules where they act faithfully and coprimely but without regular orbits, the result is new for many of the groups we consider.

The quotient of two Glauberman–Isaacs correspondences

Let a finite group A act coprimely on a finite group G. The Glauberman–Isaacs correspondence π(G, A) is a bijection from the set of A-invariant irreducible characters of G onto the set Irr(CG(A)) of irreducible characters of the centralizer of A in G. Let B be a subgroup of A. Composing from left to right, it follows that π(G, A)-1π(G, B) is an injection from Irr(CG(A)) into Irr(CG(B)). We show that, in some cases, the map can be defined via the actions of some subgroups of A containing B on the centralizers in G of some other such subgroups. We also show in many instances, such as |G| odd or A supersolvable and G solvable, that this map is independent of the overgroup G.