A solvability criterion for finite groups

We show that if for every prime p, the normalizer of a Sylow p-subgroup of a finite group G admits a p-solvable supplement, then G is solvable. This generalizes a solvability criterion of Hall which asserts that a finite group G is solvable if and only if G has a Hall p′-subgroup for every prime p.     

On the Artin exponent of some rational groups

A finite group G is called rational if all its irreducible complex characters are rational valued. In this paper, we show that if G is a direct product of finitely many rational Frobenius groups then every rationally represented character of G is a generalized permutation character. Also we show that the same assertion holds when G is a solvable rational group with a Sylow 2-subgroup isomorphic to the dihedral group of order 8 and an abelian normal Sylow 3-subgroup.     

ZEROS OF CHARACTERS ON PRIME ORDER ELEMENTS

Suppose that G is a finite group, let χ be a faithful irreducible character of degree a power of p and let P be a Sylow p-subgroup of G. If χ(x) ≠ 0 for all elements of G of order p, then P is cyclic or generalized quaternion.

     

A reduction theorem for the McKay conjecture

The McKay conjecture asserts that for every finite group G and every prime p, the number of irreducible characters of G having p’-degree is equal to the number of such characters of the normalizer of a Sylow p-subgroup of G. Although this has been confirmed for large numbers of groups, including, for example, all solvable groups and all symmetric groups, no general proof has yet been found. In this paper, we reduce the McKay conjecture to a question about simple groups. We give a list of conditions that we hope all simple groups will satisfy, and we show that the McKay conjecture will hold for a finite group G if every simple group involved in G satisfies these conditions. Also, we establish that our conditions are satisfied for the simple groups PSL₂(q) for all prime powers q≥4, and for the Suzuki groups Sz(q) and Ree groups R(q), where q=2 ^e or q=3 ^e respectively, and e>1 is odd. Since our conditions are also satisfied by the sporadic simple group J ₁, it follows that the McKay conjecture holds (for all primes p) for every finite group having an abelian Sylow 2-subgroup.     

On Two Theorems of Finite Solvable Groups

For a finite group G, let T（G） denote a set of primes such that a prime p belongs to T（G） if and only if p is a divisor of the index of some maximal subgroup of G. It is proved that if G satisfies any one of the following conditions： （1） G has a p-complement for each p∈T（G）; （2）│T（G）│= 2： （3） the normalizer of a Sylow p-subgroup of G has prime power index for each odd prime p∈T（G）; then G either is solvable or G/Sol（G）≌PSL（2, 7） where Sol（G） is the largest solvable normal subgroup of G.     

Remarks on the McKay Conjecture

The McKay Conjecture (MC) asserts the existence of a bijection between the (inequivalent) complex irreducible representations of degree coprime to p (p a prime) of a finite group G and those of the subgroup N, the normalizer of Sylow p-subgroup. In this paper we observe that MC implies the existence of analogous bijections involving various pairs of algebras, including certain crossed products, and that MC is equivalent to the analogous statement for (twisted) quantum doubles. Using standard conjectures in orbifold conformal field theory, MC is equivalent to parallel statements about holomorphic orbifolds V ^G , V ^N . There is a uniform formulation of MC covering these different situations which involves quantum dimensions of objects in pairs of ribbon fusion categories.     

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.     

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).     

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.     

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.     

An answer to a question of Isaacs on character degree graphs

Let N be a normal subgroup of a finite group G. We consider the graph Γ(G|N) whose vertices are the prime divisors of the degrees of the irreducible characters of G whose kernel does not contain N and two vertices are joined by an edge if the product of the two primes divides the degree of some of the characters of G whose kernel does not contain N. We prove that if Γ(G|N) is disconnected then G/N is solvable. This proves a strong form of a conjecture of Isaacs.     

p-Brauer characters ofq-defect 0

For ap-solvable groupG the number of irreducible Brauer characters ofG with a given vertexP is equal to the number of irreducible Brauer characters of the normalizer ofP with vertexP. In this paper we prove in addition that for solvable groups one can control the number of those characters whose degrees are divisible by the largest possibleq-power dividing the order of |G|. supported by DGICYT.PB 90-0414-C03-01     

Sylow Subgroups and Fusion of Characters

Suppose that P is a Sylow p-subgroup of G. We study irreducible characters of P that are constant on G-fused classes.     

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.     

Blocks of full defect with nonabelian metacyclic defect groups

Let p be an odd prime and let B be a p-block of a finite group G with a nonabelian metacyclic defect group P which is a Sylow p-subgroup of G. The purpose of this article is to study the ordinary and modular irreducible characters in B. In particular, we calculate k _i (B) and l _i (B) for an arbitrary nonnegative integer i.     

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.     

Finite groups and non-vanishing elements

We show that if P is a Sylow p-subgroup of G, then no irreducible character of G vanishes on any element of \({Z(P)\cap O_p(G)}\), which improves a recent result of M. Miyamoto.     

A remark on Gelfand–Graev characters for finite simple groups

The famous Gelfand–Graev character of a group of Lie type G is a multiplicity free character of shape ν ^G , where ν is a suitable degree 1 character of a Sylow p-subgroup and p is the defining characteristic of G. We show that, for an arbitrary non-abelian simple group G, if ν is a linear character of a Sylow p-subgroup of G such that ν ^G is multiplicity free, then G is isomorphic to either a group of Lie type in defining characteristic p, or to a group PSL(2, q), where either p = q + 1, or p = 2 and q + 1 or q ? 1 is a 2-power.     

A cohomological criterion for <Emphasis Type="Italic">p</Emphasis>-solvability

Let G be a finite group, let p be a prime, and let P be a Sylow p-subgroup of G. In this note we give a cohomological criterion for the p-solvability of G depending on the cohomology in degree 1 with coefficients in \(\mathbb F_p\) of both the normal subgroups of G and P. As a byproduct we bound the minimum possible number of factors of p-power order appearing in any normal series of G, in which each factor is either a p-group, a p’-group, or a non-p-solvable characteristically simple group, by the number of generators of P.