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.     

Brauer characters with cyclotomic field of values

It has been shown in an earlier paper [G. Navarro, Pham Huu Tiep, Rational Brauer characters, Math. Ann. 335 (2006) 675-686] that, for any odd prime p, every finite group of even order has a non-trivial rational-valued irreducible p-Brauer character. For p=2 this statement is no longer true. In this paper we determine the possible non-abelian composition factors of finite groups without non-trivial rational-valued irreducible 2-Brauer characters. We also prove that, if p≠q are primes, then any finite group of order divisible by q has a non-trivial irreducible p-Brauer character with values in the cyclotomic field Q(exp(2πi/q)).     

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).     

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     

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.     

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.     

Groups whose vanishing class sizes are not divisible by a given prime

Let G be a finite group. An element ${g\in G}Let G be a finite group. An element g ? G{g\in G} is a vanishing element of G if there exists an irreducible complex character χ of G such that χ(g) = 0: if this is the case, we say that the conjugacy class of g in G is a vanishing conjugacy class of G. In this paper we show that, if the size of every vanishing conjugacy class of G is not divisible by a given prime number p, then G has a normal p-complement and abelian Sylow p-subgroups.     

Correspondences Between 2-Brauer Characters of Solvable Groups

Let G be a finite solvable group and let p be a prime. Let P ∈ Syl _p (G) and N = N _G (P). We prove that there exists a natural bijection between the 2-Brauer irreducible characters of p′-degree of G and those of N _G (P).     

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     

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.     

Extensions and the Induced Characters of Cyclic p-Groups

For a prime p, we denote by B_n the cyclic group of order pⁿ. Let φ be a faithful irreducible character of B_n, where p is an odd prime. We study the p-group G containing B_n such that the induced character φ^G is also irreducible. The purpose of this article is to determine the subgroup N_G(N_G(B_n)) of G under the hypothesis [N_G(B_n):B_n]⁴ ≦ pⁿ.     

Restriction of characters and products of characters

Let G be a finite p-group, for some prime p, and ψ, θ ∈ Irr(G) be irreducible complex characters of G. It has been proved that if, in addition, ψ and θ are faithful characters, then the product ψθ is a multiple of an irreducible or it is the nontrivial linear combination of at least (p + 1)/2 distinct irreducible characters of G. We show that if we do not require the characters to be faithful, then given any integer k > 0, we can always find a p-group P and irreducible characters Ψ and Θ of P such that the product ΨΘ is the nontrivial combination of exactly k distinct irreducible characters. We do this by translating examples of decompositions of restrictions of characters into decompositions of products of characters.     

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.     

3-Blocks with Abelian defect groups isomorphic to Z_{3^m } \times Z_{3^n }

Let G be a finite group and let p be a fixed prime number. Let B be a p-block of G with defect group D. In this paper, we give results on 3-blocks with abelian defect groups isomorphic to Z3m ×Z3n. We are particularly interested in the number of irreducible ordinary characters and the number of irreducible Brauer characters in the block. We calculate two important block invariants k（B） and l（B） in this case.     

Primes dividing the degrees of the real characters

Let G be a finite group and let Irr(G) denote the set of all complex irreducible characters of G. The Ito–Michler Theorem asserts that if a prime p does not divide the degree of any χ Irr(G) then a Sylow p-subgroup P of G is normal in G. We prove a real-valued version of this theorem, where instead of Irr(G) we only consider the subset Irr_rv(G) consisting of all real-valued irreducible characters of G. We also prove that the character degree graph associated to Irr_rv(G) has at most 3 connected components. Similar results for the set of real conjugacy classes of G have also been obtained. Part of this paper was done while the second author visited the Mathematics Department of the Università di Firenze. He would like to thank the Department for its hospitality. The authors are also grateful to F. Lübeck for helping them with some computer calculations. The research of the first author was partially supported by MIUR research program “Teoria dei gruppi ed applicazioni”. This research of the second author was partially supported by the Spanish Ministerio de Educación y Ciencia proyecto MTM2004-06067-C02-01. The third author gratefully acknowledges the support of the NSA and the NSF.     

<Emphasis Type="Italic">p</Emphasis>-groups having few almost-rational irreducible characters

We prove that a 2-group has exactly five rational irreducible characters if and only if it is dihedral, semidihedral or generalized quaternion. For an arbitrary prime p, we say that an irreducible character χ of a p-group G is “almost rational” if ℚ(χ) is contained in the cyclotomic field ℚ _p , and we write ar(G) to denote the number of almost-rational irreducible characters of G. For noncyclic p-groups, the two smallest possible values for ar(G) are p ² and p ² + p − 1, and we study p-groups G for which ar(G) is one of these two numbers. If ar(G) = p ² + p − 1, we say that G is “exceptional”. We show that for exceptional groups, |G: G′| = p ², and so the assertion about 2-groups with which we began follows from this. We show also that for each prime p, there are exceptional p-groups of arbitrarily large order, and for p ≥ 5, there is a pro-p-group with the property that all of its finite homomorphic images of order at least p ³ are exceptional.     

Hamiltonicity and pancyclicity of cartesian products of graphs

The cartesian product of a graph G with K₂ is called a prism over G. We extend known conditions for hamiltonicity and pancyclicity of the prism over a graph G to the cartesian product of G with paths, cycles, cliques and general graphs. In particular we give results involving cubic graphs and almost claw-free graphs.We also prove the following: Let G and H be two connected graphs. Let both G and H have a 2-factor. If Δ(G)≤g^′(H) and Δ(H)≤g^′(G) (we denote by g^′(F) the length of a shortest cycle in a 2-factor of a graph F taken over all 2-factorization of F), then G□H is hamiltonian.     

On Thompson's Theorem

Let p be a prime divisor of the order of a finite group G. Thompson (1970, J. Algebra14, 129–134) has proved the following remarkable result: a finite group G is p-nilpotent if the degrees of all its nonlinear irreducible characters are divisible by p (in fact, in that case G is solvable). In this note, we prove that a group G, having only one nonlinear irreducible character of p′-degree is a cyclic extension of Thompson's group. This result is a consequence of the following theorem: A nonabelian simple group possesses two nonlinear irreducible characters χ₁ and χ₂ of distinct degrees such that p does not divide χ₁(1)χ₂(1) (here p is arbitrary but fixed). Our proof depends on the classification of finite simple groups. Some properties of solvable groups possessing exactly two nonlinear irreducible characters of p′-degree are proved. Some open questions are posed.     

The computations of some Schur indices

Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm _p (χ) denote the Schur index of χ overQ _p , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that for all irreducible charactersX _u ofG thenm _p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E _n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz). This work was partly supported by NSF Grant MCS-8201333.