In a recent paper, Turull showed that associated with any p-block B with cyclic defect group of any finite group are two linear characters. These two linear characters allow us to calculate easily the Hasse invariant of the characters of B. A different method to describe these Hasse invariant was proved by Nebe. In a different paper, Turull showed that the Hasse invariant of any irreducible character of a finite group G can be calculated from the Hasse invariant of the p-basic groups. In the present paper, we describe the two linear characters associated with any of the relevant characters of the p-basic groups, and we obtain in this way their Hasse invariants as well. The formulas are based on a single exceptional character of Band are fairly straightforward and uniform. (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We show that associated with each p-block B with non trivial cyclic defect group D of a finite group are two linear characters alpha Band mu B of a cyclic group E of order e. Here e is the number of non exceptional irreducible characters that B has. The first character alpha B describes the action of Eon D. The second character mu B arises from certain field extensions. The order of mu B is the p-local Schur index of any exceptional chi is an element of Irr(B), giving an alternative description of this invariant to the one given by Benard [1]. From the relationship of alpha B and mu B we give an alternative description to the one given by Nebe [6] of the invariant associated with each exceptional chi is an element of Irr(B). (c) 2025 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

We give an elementary proof of the following result. Let C be an abelian and irreducible subgroup of the symplectic group Sp (2 m , p ). Then C is cyclic and embeds in the (multiplicative) subgroup of order p^m + 1 of the field of order p^2m . The proof yields, in fact, a similar result for nonsingular bilinear forms more generally.

Let p be a prime. Let G be a finite group, and let chi be an irreducible character of G. Suppose F is a finite extension of Q(p), the field of p-adic numbers. If all the values of chi are in F, then, associated with chi, is a specific element of Br(F) the Brauer group of F, and, since Br(F) is canonically isomorphic to Q=Z, chi has, uniquely associated with it, an element of Q/Z: Some recent conjectures and results on modular representation theory depend on the value of this element of Br(F). When chi takes complex values, the element of Br(F) associated with chi is not, in general, uniquely determined. In this paper, we show that, whenever p-Brauer characters are defined, then the element of Br(F) associated with chi can be naturally and uniquely defined.

Let K be a finite extension of the field Qp of p-adic numbers. Let χ be an irreducible character of a finite group whose values are in K. Associated with χ and K is an element of the Brauer group of K, and therefore by standard results a local invariant in Q/Z. The paper defines some classes of finite groups, that the paper calls p-basic groups, and gives formulas to calculate the local invariant of their characters. It also shows how one can calculate the local invariant of any irreducible character of any finite group that has values in K by reducing the problem to the case where the groups are p-basic and using the explicit formulas given in the paper.

We prove the following result. Let p be a prime, and let G be a finite p-solvable group that is a direct product of two non-cyclic subgroups of coprime order, and let V be some faithful irreducible module for G over some field in characteristic p. Then G has a regular orbit, that is, there exists some v∈V such that CG(v)=1.

Dade's Projective Conjecture is known to be true for finite p-solvable groups thanks to work of G.R. Robinson, but remains open in general. Work of Isaacs and Navarro suggested to Uno and Boltje refinements of this conjecture. These refinements were studied for finite p-solvable groups by Glesser. In the present paper, inspired by earlier work of Turull, we propose further refinements of the conjecture that take into account the Schur indices and the elements of the Brauer group. We prove that all these refinements of Dade's Projective Conjecture hold for all finite p-solvable groups. In particular, we obtain that the version of Dade's Projective Conjecture which involves character degree residues modulo p, fields of definition and Schur indices, as well as the full strength of Boltje's Conjecture both hold for all finite p-solvable groups. The proof develops a Clifford theory for normalizers of chains of p-subgroups which allows one to reduce the calculation of the relevant sums to simpler groups.

This paper concerns aspects of Clifford Theory of finite groups. In earlier papers, Turull proved that if two finite groups yielded the same element of the Brauer–Clifford group, then there was an endoisomorphism from one group to the other, and furthermore, that associated with each endoisomorphism there was an essentially unique correspondence of modules over many different fields from one group to the other. The paper adapts the definition of Character Triple Isomorphism so that it involves ordinary and Brauer characters, and it preserves fields of definition, Schur indices, decomposition numbers, and blocks. It is proved that each endoisomorphism yields exactly one character triple isomorphism. Character triple isomorphisms can be composed, restricted, produced by direct sums, extension of fields, and these operations have their parallel for the endoisomorphisms. One goal of the paper is to provide tools for the study of the character theory of finite groups in an accessible way suitable for applications.

Let A be a finite group acting on a finite group G with (vertical bar A vertical bar; vertical bar G vertical bar) = 1. The Glauberman-Isaacs correspondence is often viewed as providing for each pair of subgroups B-1 (sic) B-2 of A a bijection Irr(B2)(CG(B-1)) -> Irr(CG(B)(2))), where Irr(B2)(CG(B-1)) is the set of all irreducible characters of CG(B-1) which are invariant under B-2. The inverse Glauberman- Isaacs correspondence can be viewed as providing an injective map IrrCG(B-2)) -> IrrCG(B-1). We show that this can naturally be extended to give a corresponding map for each B-1 a subnormal subgroup of B-2, where B-2 is any subgroup of A. The properties of this correspondence are very close to those of the original one.

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

Let C be a set of finite groups which is closed under taking subgroups and let d and M be positive integers. Suppose that for every G∈C whose order is divisible by at most two distinct primes there exists an abelian subgroup A⊆G such that A is generated by d or fewer elements and [G:A]≤M. We prove that there exists a positive constant C0 such that every G∈C has an abelian subgroup A satisfying [G:A]≤C0, and A can be generated by d or fewer elements. We also prove some related results. Our proofs use the Classification of Finite Simple Groups.

The Alperin Weight Conjecture is a well-known conjecture, and it is central to the modern representation theory of finite groups. It is known to be true for many types of finite groups, but it remains open. In particular, the conjecture is known to be true for finite p-solvable groups. In this paper, we prove that a very strong form of the conjecture holds for all finite p-solvable groups. It follows that the strengthening of the Alperin Weight Conjecture that includes Galois automorphisms over the p-adic numbers holds for all finite p-solvable groups. This strengthening of the Alperin Weight Conjecture was suggested in an earlier work of Navarro on the McKay Conjecture. The present paper provides the first direct evidence (beyond the calculation of individual cases and the case of groups of odd order) for this strengthening.

Let G be a finite nilpotent group. Suppose that G 0 is a subgroup of G and that ψ is an irreducible character of G 0 . Consider the set S whose elements are the natural numbers m_ Q(χ)[ Q(χ) : Q] as χ runs through the irreducible characters of G which contain ψ as a summand when restricted to G 0 . Here m Q (χ) is, as usual, the rational Schur index of χ , and [ Q(χ) : Q] is the degree of the extension of the field of values of the character as an extension of the rationals. We prove that then the minimum element of S divides all the other elements of S . The result is not true when G is an arbitrary finite group. We also consider some variations of this result.

The Alperin–McKay conjecture is a well-known conjecture. It is known to be true for p-solvable groups by work of Dade and Okuyama–Wajima. Recently, this conjecture has been strengthened by work of Isaacs–Navarro, Navarro and Turull. This refinement involves the degrees modulo p of the characters involved, the field of values over the p-adic numbers of the relevant characters, and their p-local Schur indices. In this paper, we prove that this strengthened version of the conjecture is true for all p-solvable groups.

In an earlier paper (Turull, 2012 [12]) the author introduced the concept of endoisomorphisms and showed its natural connection with Clifford theory of finite groups. Associated with Clifford theory are module and character correspondences. In the present paper the author shows that each endoisomorphism produces a unique module and character correspondence with excellent compatibility results.

This paper describes the relation between Clifford theory of finite groups over any field, and the Brauer–Clifford group. Let G and G¯ be finite groups, let π:G→G¯ be a surjective group homomorphism with kernel H, and let F be any field. Let K be any extension field of F and let S be an irreducible KH-module. We show that to S is associated in a natural way a specific element [[S,π,F]] of a Brauer–Clifford group defined over π and F. As a tool to prove the existence of this association, and to study its properties, we use endoisomorphisms. These are simply certain isomorphisms of related endomorphism algebras as G¯-algebras over F. We show that two modules for two different finite groups have the same (in an appropriate sense) element of the Brauer–Clifford group if and only if there exists an appropriate endoisomorphism.

The Brauer–Clifford group was introduced to describe the Clifford theory for finite groups. It was proved that it has a natural homomorphism into a Brauer group, and the kernel of this homomorphism is the set of all equivalence classes of G-algebras which are full matrix algebras. In this paper, we prove that this kernel is isomorphic to a second cohomology group. In the Clifford theory for finite groups situation, we characterize families of characters which yield elements in the full matrix subgroup of the Brauer–Clifford group as those where an appropriate character has Schur index one. We also show, in this case, how to compute the element of the second cohomology group associated with this family of characters.

In previous work, the author introduced the Brauer–Clifford group of certain G-algebras. This group is useful because to every irreducible character of a normal subgroup of a finite group, one can associate a unique element of a specific Brauer–Clifford group, and this element controls the Clifford theory of this character in its ambient group. In the present paper, we define the Brauer–Clifford group of G-rings. This new definition only requires us to discuss tensor products over the underlying G-ring, and it is simpler than the earlier one. We prove that the new definition yields a group which is canonically isomorphic to the Brauer–Clifford group of a corresponding suitable G-algebra.

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

Let S be a finite solvable group, and suppose S acts on the finite group N, and they have coprime orders. Then, the celebrated Glauberman correspondence provides a natural bijection from the set IrrS(N) of irreducible characters of N which are invariant under the action of S to the set Irr(CN(S)) of all irreducible characters of the centralizer of S in N. Suppose, further, that the semidirect product SN is a normal subgroup of a finite group G. Let θ∈IrrS(N), and let ψ∈Irr(CN(S)) be its Glauberman correspondent. We prove that there is a bijection with good compatibility properties from the set Irr(G,θ) of the irreducible characters of G above θ to Irr(NG(S),ψ) such that, in the case when S is a p-group for some prime p, it preserves fields of values and Schur indices over Qp, the field of p-adic numbers. Using this result, we also prove a strengthening of the McKay Conjecture for all p-solvable groups.