Rational Brauer characters

It is known that every finite group of even order has a non-trivial complex irreducible character which is rational valued. We prove the modular version of this result: If p is an odd prime and G is any finite group of even order, then G has a non-trivial irreducible p-Brauer character which is rational valued. The first author is partially supported by the Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01, while the second gratefully acknowledges the support of the NSA (grant H98230-04-0066).     

Rational irreducible characters and rational conjugacy classes in finite groups

We prove that a finite group has two rational-valued irreducible characters if and only if it has two rational conjugacy classes, and determine the structure of any such group. Along the way we also prove a conjecture of Gow stating that any finite group of even order has a non-trivial rational-valued irreducible character of odd degree.

     

Brauer characters and rationality

A finite group $G$ has no non-trivial rational-valued irreducible $p$ -Brauer characters if and only if $G$ has no non-trivial rational elements of order not divisible by $p$ .     

Finite groups whose irreducible Brauer characters have prime power degrees

Let G be a finite group and let p be a prime. In this paper, we classify all finite quasisimple groups in which the degrees of all irreducible p-Brauer characters are prime powers. As an application, for a fixed odd prime p, we classify all finite nonsolvable groups with the above-mentioned property and having no nontrivial normal p-subgroups. Furthermore, for an arbitrary prime p, a complete classification of finite groups in which the degrees of all nonlinear irreducible p-Brauer characters are primes is also obtained.     

Zeros of Brauer characters

The authors obtain a sufficient condition to determine whether an element is a vanishing regular element of some Brauer character. More precisely, let G be a finite group and p be a fixed prime, and H = G′ O^p′ (G); if g ∈ G⁰ - H⁰ with o(gH) coprime to the number of irreducible p-Brauer characters of G, then there always exists a nonlinear irreducible p-Brauer character which vanishes on g. The authors also showin this note that the sums of certain irreducible p-Brauer characters take the value zero on every element of G⁰ - H⁰.     

A note on degrees of irreducible π-Brauer characters

In this note, we first investigate the degrees of irreducible π-Brauer characters of a finite π-separable group G. Then we relate degrees of irreducible π-Brauer characters to class lengths of π-regular elements of G. Received: 4 April 2008     

Degrees of rational characters of finite groups

A classical theorem of John Thompson on character degrees states that if the degree of any complex irreducible character of a finite group G is 1 or divisible by a prime p, then G has a normal p-complement. In this paper, we consider fields of values of characters and prove some improvements of this result.     

Odd character correspondences in solvable groups

Let G be a finite solvable group, p be some prime, let P be a Sylow p-subgroup of G, and let N be its normalizer in G. Assume that N has odd order. Then, we prove that there exists a bijection from the set of all irreducible characters of G of degree prime to p to the set of all the irreducible characters of degree prime to p of N such that it preserves ± the degree modulo p, the field of values, and the Schur index over every field of characteristic zero. This strengthens a more general recent result [A. Turull, Character correspondences in solvable groups, J. Algebra 295 (2006) 157–178], but only for the case under consideration here. In addition, we prove some other strong character correspondences that have very good rationality properties. As one consequence, we prove that a solvable group G has a non-trivial rational irreducible character with degree prime to p if and only if the order of the normalizer of a Sylow p-subgroup of G has even order.     

Representations of finite special linear groups in non-defining characteristic

We classify irreducible modules over the finite special linear group SLn(q) in the non-defining characteristic ?, describe restrictions of irreducible modules from GLn(q) to SLn(q), classify complex irreducible characters of SLn(q) irreducible modulo l, and discuss unitriangularity of the l-decomposition matrix for SLn(q).     

Irreducible <Emphasis Type="Italic">p</Emphasis>-Brauer characters of <Emphasis Type="Italic">p</Emphasis>-power degree for the symmetric and alternating groups

Starting from the question when all irreducible p-Brauer characters for a symmetric or an alternating group are of p-power degree, we classify the p-modular irreducible representations of p-power dimension in some families of representations for these groups. In particular, this then allows to confirm a conjecture by W. Willems for the alternating groups. Received: 14 June 2006     

Weak Mp-groups

Abstract

Let G be a finite group and p be a prime. We prove in this note that if every irreducible monolithic p-Brauer character of G is monomial then G is solvable.

Communicated by J. Zhang     

Taketa’s theorem for some character degree sets

Let cd(G) be the set of irreducible complex character degrees of a finite group G. The Taketa problem conjectures that if G is a finite solvable group, then ${{\rm dl}(G) \leqslant |{\rm cd} (G)|}$ , where dl(G) is the derived length of G. In this note, we show that this inequality holds if either all nonlinear irreducible characters of G have even degrees or all irreducible character degrees are odd. Also, we prove that this inequality holds if all irreducible character degrees have exactly the same prime divisors. Finally, Isaacs and Knutson have conjectured that the Taketa problem might be true in a more general setting. In particular, they conjecture that the inequality ${{\rm dl}(N) \leqslant |{\rm cd} {(G \mid N)}|}$ holds for all normal solvable subgroups N of a group G. We show that this conjecture holds if ${{\rm cd} {(G \mid N')}}$ is a set of non-trivial p–powers for some fixed prime p.     

A new characterization for some extensions of PSL(2, <Emphasis Type="Italic">q)</Emphasis> for some <Emphasis Type="Italic">q</Emphasis> by some character degrees

In [22] (Tong-Viet H P, Simple classical groups of Lie type are determined by their character degrees, J. Algebra, 357 (2012) 61–68), the following question arose: Which groups can be uniquely determined by the structure of their complex group algebras? The authors in [12] (Khosravi B et al., Some extensions of PSL(2,p²) are uniquely determined by their complex group algebras, Comm. Algebra, 43(8) (2015) 3330–3341) proved that each extension of PSL(2,p²) of order 2|PSL(2,p²)| is uniquely determined by its complex group algebra. In this paper we continue this work. Let p be an odd prime number and q = p or q = p³. Let M be a finite group such that |M| = h|PSL(2,q), where h is a divisor of |Out(PSL(2,q))|. Also suppose that M has an irreducible character of degree q and 2p does not divide the degree of any irreducible character of M. As the main result of this paper we prove that M has a unique nonabelian composition factor which is isomorphic to PSL(2,q). As a consequence of our result we prove that M is uniquely determined by its order and some information on its character degrees which implies that M is uniquely determined by the structure of its complex group algebra.     

Character tables of p-groups with derived subgroup of prime order II

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. In this paper we classify up to isomorphism the character tables of p-groups with derived subgroup of prime order via a combinatorial object independent of the prime p.     

Cellular Structure of q-Brauer Algebras

Let R be a commutative noetherian domain. The q-Brauer algebras over R are shown to be cellular algebras in the sense of Graham and Lehrer. In particular, they are iterated inflations of Hecke algebras of type A. When R is a field of arbitrary characteristic, we determine for which parameters the q-Brauer algebras are quasi-hereditary. Then, using the general theory of cellular algebras we parametrize all irreducible representations of q-Brauer algebras.     

Block separations and inclusions

We investigate the separation of characters by blocks at different primes and the inclusions of q-blocks in p-blocks (viewed as sets of characters), and use these notions to prove results on the structure of the corresponding groups. In particular, we provide a new criterion for the nilpotence of a finite group G based on the separation by principal blocks, and we show that a condition on block unions has strong structural consequences.     

Groups Satisfying the Two-Prime Hypothesis with a Composition Factor Isomorphic to PSL<Subscript>2</Subscript>(<Emphasis Type="Italic">q</Emphasis>) for <Emphasis Type="Italic">q</Emphasis> ⩾ 7

Let G be a finite group and write cd(G) for the degree set of the complex irreducible characters of G. The group G is said to satisfy the two-prime hypothesis if for any distinct degrees a, b 2 cd(G), the total number of (not necessarily different) primes of the greatest common divisor gcd(a, b) is at most 2. We prove an upper bound on the number of irreducible character degrees of a nonsolvable group that has a composition factor isomorphic to PSL₂(q) for q ? 7.     

Character tables of p-groups with derived subgroup of prime order I

Two character tables of finite groups are isomorphic if there exist a bijection for the irreducible characters and a bijection for the conjugacy classes that preserve all the character values. We give necessary and sufficient conditions for two finite groups to have isomorphic character tables. In the case of finite p-groups with derived subgroup of order p, we show that the character tables can be classified by equivalence classes of certain homomorphisms of abelian p-groups.     

Above the Glauberman correspondence

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.     

Counting characters of small degree in upper unitriangular groups

Let U_n denote the group of upper n×n unitriangular matrices over a fixed finite field of order q. That is, U_n consists of upper triangular n×n matrices having every diagonal entry equal to 1. It is known that the degrees of all irreducible complex characters of U_n are powers of q. It was conjectured by Lehrer that the number of irreducible characters of U_n of degree q^e is an integer polynomial in q depending only on e and n. We show that there exist recursive (for n) formulas that this number satisfies when e is one of 1,2 and 3, and thus show that the conjecture is true in those cases.