P-subgroup Research Articles

Constructing characters of Sylow p-subgroups of finite Chevalley groups

Let q be a power of a prime p, let G be a finite Chevalley group over Fq and let U be a Sylow p-subgroup of G; we assume that p is not a very bad prime for G. We explain a procedure of reduction of irreducible complex characters of U, which leads to an algorithm whose goal is to obtain a parametrization of the irreducible characters of U along with a means to construct these characters as induced characters. A focus in this paper is determining the parametrization when G is of type F4, where we observe that the parametrization is “uniform” over good primes p>3, but differs for the bad prime p=3. We also explain how it has been applied for all groups of rank 4 or less.

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.

Irreducible characters of even degree and normal Sylow 2-subgroups

AbstractThe classical Itô-Michler theorem on character degrees of finite groups asserts that if the degree of every complex irreducible character of a finite group G is coprime to a given prime p, then G has a normal Sylow p-subgroup. We propose a new direction to generalize this theorem by introducing an invariant concerning character degrees. We show that if the average degree of linear and even-degree irreducible characters of G is less than 4/3 then G has a normal Sylow 2-subgroup, as well as corresponding analogues for real-valued characters and strongly real characters. These results improve on several earlier results concerning the Itô-Michler theorem.

Character degree graphs of solvable groups with diameter three

Abstract Let G be a finite solvable group and cd ⁡ ( G ) ${\operatorname{cd}(G)}$ the set of character degrees of G. The character degree graph Δ ⁢ ( G ) ${\Delta(G)}$ is the graph whose vertices are the primes dividing the degrees in cd ⁡ ( G ) ${\operatorname{cd}(G)}$ and there is an edge between two distinct primes p and q if their product pq divides some degree in cd ⁡ ( G ) ${\operatorname{cd}(G)}$ . When Δ ⁢ ( G ) ${\Delta(G)}$ has diameter three, we can partition the vertices ρ ⁢ ( G ) ${\rho(G)}$ into four non-empty disjoint subsets ρ 1 ∪ ρ 2 ∪ ρ 3 ∪ ρ 4 ${\rho_{1}\cup\rho_{2}\cup\rho_{3}\cup\rho_{4}}$ where no prime in ρ 1 ${\rho_{1}}$ is adjacent to any prime in ρ 3 ∪ ρ 4 ${\rho_{3}\cup\rho_{4}}$ ; no prime in ρ 4 ${\rho_{4}}$ is adjacent to any prime in ρ 1 ∪ ρ 2 ${\rho_{1}\cup\rho_{2}}$ ; every prime in ρ 2 ${\rho_{2}}$ is adjacent to some prime in ρ 3 ${\rho_{3}}$ ; every prime in ρ 3 ${\rho_{3}}$ is adjacent to some prime in ρ 2 ${\rho_{2}}$ ; and | ρ 1 ∪ ρ 2 | ≤ | ρ 3 ∪ ρ 4 | ${|\rho_{1}\cup\rho_{2}|\leq|\rho_{3}\cup\rho_{4}|}$ . We will show the following: If G is a solvable group where Δ ⁢ ( G ) ${\Delta(G)}$ has diameter three, then ρ 3 ${\rho_{3}}$ has at least three vertices and G has a normal non-abelian Sylow p-subgroup where p ∈ ρ 3 ${p\in\rho_{3}}$ . If ρ 1 ∪ ρ 2 ${\rho_{1}\cup\rho_{2}}$ has n vertices, then ρ 3 ∪ ρ 4 ${\rho_{3}\cup\rho_{4}}$ must have at least 2 n ${2^{n}}$ vertices. The group G has Fitting height 3.

A generalization of Burnside’s p-nilpotency criterion

Abstract Let p be a prime number, let G be a finite group, and let P be a Sylow p-subgroup of G. Under the assumption that N G ⁢ ( P ) ${N_{G}(P)}$ is p-nilpotent, we give an equivalent condition of the p-nilpotency of G. Our result is a generalization of the famous Burnside p-nilpotency criterion and some recent results.

Equalities on the number of conjugacy classes of the Sylow p-subgroups of GL(n, q)

We denote by Gn the group of the upper unitriangular matrices over Fq, the finite field with q = pt elements, and r(Gn) the number of conjugacy classes of Gn. In this paper, we obtain the value of r(Gn) modulo (q2 -1)(q -1). We prove the following equalities

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.

Local c*-Supplementation of Some Subgroups in Finite Groups

A subgroup H of a finite group G is called c*-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is S-quasinormally embedded in G. In this paper, we investigate the local c*-supplementation of maximal subgroups of some Sylow p-subgroup and present some sufficient and necessary conditions for a finite group to be p-nilpotent. As applications, we give some sufficient conditions for a finite group to be in a saturated formation.

HOMOLOGICAL DIMENSIONS OF CROSSED PRODUCTS

AbstractIn this paper, we consider several homological dimensions of crossed products AασG, where A is a left Noetherian ring and G is a finite group. We revisit the induction and restriction functors in derived categories, generalizing a few classical results for separable extensions. The global dimension and finitistic dimension of AσαG are classified: global dimension of AσαG is either infinity or equal to that of A, and finitistic dimension of AσαG coincides with that of A. A criterion for skew group rings to have finite global dimensions is deduced. Under the hypothesis that A is a semiprimary algebra containing a complete set of primitive orthogonal idempotents closed under the action of a Sylow p-subgroup S ≤ G, we show that A and AασG share the same homological dimensions under extra assumptions, extending the main results in (Li, Representations of modular skew group algebras, Trans. Amer. Math. Soc.367(9) (2015), 6293–6314, Li, Finitistic dimensions and picewise hereditary property of skew group algebras, to Glasgow Math. J.57(3) (2015), 509–517).

Products of Sylow Subgroups in Suzuki and Ree Groups

An explicit and elementary proof is given to the fact that Suzuki and Ree groups can be decomposed into the product of 4 of their Sylow p-subgroups, where p is the defining characteristic.

Non-vanishing elements in finite groups

Many results have been established about determining whether or not an element evaluates to zero on an irreducible character of a group. In this note it is shown that if a group G has a normal p-subgroup N, and x∈N lies in the centre of a Sylow p-subgroup of G, then no irreducible character of G vanishes on x.

Principal blocks and p-radical groups

Let G be a finite group and k a field of characteristic p > 0. In this paper, we obtain several equivalent conditions to determine whether the principal block B 0 of a finite p-solvable group G is p-radical, which means that B 0 has the property that e 0(k P ) G is semisimple as a kG-module, where P is a Sylow p-subgroup of G, k P is the trivial kP-module, (k P ) G is the induced module, and e 0 is the block idempotent of B 0. We also give the complete classification of a finite p-solvable group G which has not more than three simple B 0-modules where B 0 is p-radical.

Large $${\varvec{p}}^{\prime }$$ p ′ -orbits for $${\varvec{p}}$$ p -nilpotent linear groups

Let G be a p-nilpotent linear group on a finite vector space V of characteristic p. Suppose that |G||V| is odd. Let P be a Sylow p-subgroup of G. We show that there exist vectors \(v_1\) and \(v_2\) in V such that \(C_G(v_1) \cap C_G(v_2)=P\). A striking conjecture of Malle and Navarro offers a simple global criterion for the nilpotence (in the sense of Broue and Puig) of a p-block of a finite group. Our result implies that this conjecture holds for groups of odd order.

Finite groups with given systems of weakly S-propermutable subgroups

Abstract A subgroup H of a finite group G is said to be S-propermutable in G provided there is a subgroup B of G such that G = N G ⁢ ( H ) ⁢ B ${G=N_{G}(H)B}$ and H permutes with all Sylow subgroups of B (see [18]). In this paper, we study the following generalization of the concept of S-permutability: A subgroup H of a finite group G is called weakly S-propermutable in G provided there is a K- 𝔘 ${\mathfrak{U}}$ -subnormal subgroup (see [1, p. 236]) T of G such that H ⁢ T = G ${HT=G}$ and H ∩ T ≤ S ≤ H ${H\cap T\leq S\leq H}$ for some S-propermutable subgroup S of G. Our main goal here is to prove the following result (Theorem 1.2): Let X be a normal subgroup of G and P a Sylow p-subgroup of X, where ( p - 1 , | X | ) = 1 ${(p-1,|X|)=1}$ . Suppose that there is a subgroup 1 < D < P ${1<D<P}$ such that every subgroup of P of order | D | ${|D|}$ and also, in the case when P is non-abelian and | D | = 2 ${|D|=2}$ , every cyclic subgroup of order 4 of P is weakly S-propermutable in G. Then X is p-nilpotent and every p-chief factor of G below X is cyclic. This theorem generalizes many known results and also gives the positive answer to Question 18.91 (b) in Kourovka Notebook [14].

Odd-Degree Characters and Self-Normalizing Sylow 2-Subgroups: A Reduction to Simple Groups

Let G be a a finite group, p a prime, and P a Sylow p-subgroup of G. A recent refinement, due to G. Navarro, of the McKay conjecture suggests that there should exist a bijection between irreducible characters of p′-degree of G and NG(P) which commutes with certain Galois automorphisms. In this article, we explore one of the consequences of this refinement, namely a way to read off from the character table of G whether a Sylow 2-subgroup of G is self-normalizing. We provide a reduction to finite simple groups and begin an investigation of the problem for simple groups.

A note on a result of Guo and Isaacs about p-supersolubility of finite groups

In this note, global information about a finite group is obtained by assuming that certain subgroups of some given order are S-semipermutable. Recall that a subgroup H of a finite group G is said to be S-semipermutable if H permutes with all Sylow subgroups of G of order coprime to \({\lvert H\rvert}\). We prove that for a fixed prime p, a given Sylow p-subgroup P of a finite group G, and a power d of p dividing \({\lvert G\rvert}\) such that \({1\le d d}\). This extends the main result of Guo and Isaacs in (Arch. Math. 105:215–222 2015). We derive some theorems that extend some known results concerning S-semipermutable subgroups.

Sylow subgroups of symmetric and alternating groups and the vertex of [formula omitted] in characteristic p

We show that the Sylow p-subgroups of a symmetric group, respectively an alternating group, are characterized as the p-subgroups containing all elementary abelian p-subgroups up to conjugacy of the symmetric group, respectively the alternating group. We apply the characterization result for symmetric groups to compute the vertices of the hook Specht modules associated to the partition (kp−p,1p) under the assumption that k≡1modp and k≢1modp2.

Finite groups with only one p-singular Brauer character degree

Recently, Isaacs, Moretó, Navarro, and Tiep investigated finite groups with just one irreducible character degree divisible by a given prime p, and showed that their Sylow p-subgroups are almost normal and almost abelian. In this paper, we consider the corresponding situation for Brauer characters. In particular, we show that if a finite group G has just one irreducible p-Brauer character degree n divisible by p≥5 then either G/Op(G) has a non-normal T.I. Sylow p-subgroup of order np, or G has a nonabelian chief factor of order divisible by p that is unique and is a simple group of Lie type of characteristic p.

Finite groups with generalized subnormal embedding of Sylow subgroups

Given a set π of primes and a hereditary saturated formation F, we study the properties of the class of groups G for which the identity subgroup and all Sylow p-subgroups are F-subnormal (K-F-subnormal) in G for each p in π. We show that such a class is a hereditary saturated formation and find its maximal inner local screen. Some criteria are obtained for the membership of a group in a hereditary saturated formation in terms of its formation subnormal Sylow subgroups.

The invariant rings of the Sylow groups of GU(3,q2), GU(4,q2), Sp(4,q) and O+(4,q) in the natural characteristic

Let G be a Sylow p-subgroup of the unitary groups GU(3,q2), GU(4,q2), the symplectic group Sp(4,q) and, for q odd, the orthogonal group O+(4,q). In this paper we construct a presentation for the invariant ring of G acting on the natural module. In particular we prove that these rings are generated by orbit products of variables and certain invariant polynomials which are images under Steenrod operations, applied to the respective invariant form defining the corresponding classical group. We also show that these generators form a SAGBI basis and the invariant ring for G is a complete intersection.