P-solvable Groups Research Articles

Inertial blocks of finite groups over arbitrary fields

We define inertial blocks of finite groups over arbitrary fields and prove that a block covered by an inertial block over a bigger field of characteristic p is still inertial if p is odd or if the inertial block has abelian defect groups. Combined with some known facts, this result implies that Broué's conjecture over arbitrary fields holds for many blocks, for example, blocks for p-solvable groups with abelian defect groups. Consequently, we partially prove Puig's finiteness conjecture for inertial blocks over fields with odd characteristic or with abelian defect groups.

Primes and conductors of characters

If G is a finite p-solvable group, P is a Sylow p-subgroup of G and χ∈Irr(G) is an irreducible complex character of degree not divisible by p, we prove that the field of values Q(χP) of the restriction of χ to P has index not divisible by p in the cyclotomic extension Q(e2πipf), where pf is the p-part of the conductor of χ. This proves the p-solvable case of the main conjecture in [4].

On the degrees of irreducible characters fixed by some field automorphism in 𝑝-solvable groups

It is known that, if all the real-valued irreducible characters of a finite group have odd degree, then the group has normal Sylow 2 2 -subgroup. This result is generalized for Sylow p p -subgroups, for any prime number p p , while assuming the group to be p p -solvable. In particular, it is proved that a p p -solvable group has a normal Sylow p p -subgroup if p p does not divide the degree of any irreducible character of the group fixed by a field automorphism of order p p .

Lifts of Brauer characters in characteristic two

A conjecture raised by Cossey in 2007 asserts that if G is a finite p-solvable group and φ is an irreducible p-Brauer character of G with vertex Q, then the number of lifts of φ is at most |Q:Q′|. This conjecture is now known to be true in several situations for p odd, but there has been little progress for p even. The main obstacle appeared in characteristic two is that all the vertex pairs of a lift might be neither linear nor conjugate. In this paper we show that if χ is a lift of an irreducible 2-Brauer character in a solvable group, then χ has a linear Navarro vertex if and only if all the vertex pairs of χ are linear, and in that case all of the twisted vertices of χ are conjugate. Our result can also be used to study other lifting problems of Brauer characters in characteristic two. As an application, we prove a weaker form of Cossey's conjecture for p=2 “one vertex at a time”.

Navarro vertices and lifts in solvable groups

Let Q be a p-subgroup of a finite p-solvable group G, where p is a prime, and suppose that δ is a linear character of Q with the property that whenever are conjugate in G. In this situation, we show that restriction to p-regular elements defines a canonical bijection from the set of those irreducible ordinary characters of G with Navarro vertex onto the set of irreducible Brauer characters of G with Green vertex Q. Also, we use this correspondence to examine the behavior of lifts of Brauer characters with respect to normal subgroups. Communicated by Mandi Schaeffer Fry

Characters, exponents and defects in p-solvable groups

Characters, exponents and defects in p-solvable groups

Character Triple Conjecture for p-solvable groups

In this paper, we prove the Character Triple Conjecture for p-solvable groups. This is a conjecture proposed by Späth during the reduction process of Dade's Projective Conjecture to quasisimple groups (see [29]). In addition, as suggested by Isaacs and Navarro in [13], we take into account the p-residue of characters.

Notes on normal p-complements and monomial characters

Let G be a finite group and Irr(G) be the set of irreducible (complex) characters of G. Let χ ∈ Irr(G) and write cod(χ) = |G : ker χ|/χ(1) as the codegree of χ, where ker χ is the kernel of χ. Denote by the intersection of the kernels of all the monolithic, monomial irreducible characters of a p-solvable group G with codegrees divisible by a prime p. We prove in this note that has a normal p-complement, and then the theorem in [2] is generalized. Also, we prove that about the theorem in [2] more is true.

Restrictions of characters in p-solvable groups

Let G be a p-solvable group, P≤G a p-subgroup and χ∈Irr(G) such that χ(1)p≥|G:P|p. We prove that the restriction χP is a sum of characters induced from subgroups Q≤P such that χ(1)p=|G:Q|p. This generalizes previous results by Giannelli–Navarro and Giannelli–Sambale on the number of linear constituents of χP. Although this statement does not hold for arbitrary groups, we conjecture a weaker version which can be seen as an extension of Brauer–Nesbitt's theorem on characters of p-defect zero. It also extends a conjecture of Wilde.

Linear characters of Sylow subgroups of symmetric groups

Let p be any prime. Let Pn be a Sylow p-subgroup of the symmetric group Sn. Let ϕ and ψ be linear characters of Pn and let N be the normaliser of Pn in Sn. In this article we show that the inductions of ϕ and ψ to Sn are equal if, and only if, ϕ and ψ are N–conjugate. This is an analogue for symmetric groups of a result of Navarro for p-solvable groups.

The semi p-cover-avoidance properties of p-sylowizers in finite groups

A subgroup S of G is called a p-sylowizer of a p-subgroup R in G if S is maximal in G with respect to having R as its Sylow p-subgroup. In this paper, we give some new characterizations of p-solvable and p-supersolvable groups by the p-cover-avoidance properties of the p-sylowizers of some p-subgroups and the p-sylowizers in some classes of groups.

Generalized bases of finite groups

Motivated by recent results on the minimal base of a permutation group, we introduce a new local invariant attached to arbitrary finite groups. More precisely, a subset Delta of a finite group G is called a p-base (where p is a prime) if langle Delta rangle is a p-group and mathrm {C}_G(Delta ) is p-nilpotent. Building on results of Halasi–Maróti, we prove that p-solvable groups possess p-bases of size 3 for every prime p. For other prominent groups, we exhibit p-bases of size 2. In fact, we conjecture the existence of p-bases of size 2 for every finite group. Finally, the notion of p-bases is generalized to blocks and fusion systems.

P-parts of co-degrees of irreducible characters

For a character χ of a finite group G, the co-degree of χ is χ c (1)=[G:kerχ] χ(1). Let p be a prime and let e be a positive integer. In this paper, we first show that if G is a p-solvable group such that p e+1 ∤χ c (1), for every irreducible character χ of G, then the p-length of G is not greater than e. Next, we study the finite groups satisfying the condition that p 2 does not divide the co-degrees of their irreducible characters.

On class sizes and primitive character degrees of p-solvable groups

We improve a result of Isaacs et al. with respect to character degrees and class sizes of finite groups [1, Corollary D].

Coprime actions with p-nilpotent centralizers

Let A be an elementary abelian group of order r2 acting coprimely on a finite p-solvable group G. We prove that if CG(a) is p-nilpotent for each non-trivial element a in A, then G is meta-p-nilpotent, i.e. p-nilpotent-by-p-nilpotent. In fact, our results are more general in this paper.

Some conjectures on Brauer character degrees

Let G be a finite group and let p be a prime. Willems conjectured that the sum of the squares of all Brauer character degrees of a finite group G is bounded below by the p′-part of the order of G. In this note, we propose a ‘projective’ version of this conjecture and prove it for p-solvable groups. Using this, we manage to reduce the proof of Willems conjecture to a question concerning quasi-simple groups. We also introduce a projective version of a conjecture due to Martínez-Pérez and Willems on projective indecomposable characters which, if true, implies the veracity of the projective version of the Willems conjecture above.

A note on vertices of indecomposable tensor products

Abstract G. Navarro raised the question of when two vertices of two indecomposable modules over a finite group algebra generate a Sylow p-subgroup. The present note provides a sufficient criterion for this to happen. This generalises a result by Navarro for simple modules over finite p-solvable groups, which is the main motivation for this note.

P-length and character degrees in p-solvable groups

Let G be a p-solvable group, where p is a prime. We prove that the p-length of G is less or equal then the number of distinct irreducible character degrees of G not divisible by p. Furthermore, we prove that the result still holds if we impose some restriction on the field of values of the characters. In particular, if p=2, we can consider only rational-valued characters.

Remarks on Harada’s conjecture

An open conjecture by Harada from 1981 gives an easy characterization of the p-blocks of a finite group in terms of the ordinary character table. Kiyota and Okuyama have shown that the conjecture holds for p-solvable groups. In the present work we extend this result using a criterion on the decomposition matrix. In this way we prove Harada’s Conjecture for several new families of defect groups and for all blocks of sporadic simple groups. In the second part of the paper we present a dual approach to Harada’s Conjecture.

On Quillen's conjecture for p-solvable groups

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without employing it.