Frobenius Action on Carter Subgroups (original) (raw)

Action of a Frobenius-like group

Journal of Algebra, 2014

We call a finite group Frobenius-like if it has a nontrivial nilpotent normal subgroup F possessing a nontrivial complement H such that [F, h] = F for all nonidentity elements h ∈ H. We prove that any irreducible nontrivial F Hmodule for a Frobenius-like group F H of odd order over an algebraically closed field has an H-regular direct summand if either F is fixed point free on V or F acts nontrivially on V and the characteristic of the field is coprime to the order of F. Some consequences of this result are also derived.

Action of a Frobenius-like group with kernel having central derived subgroup

International Journal of Algebra and Computation, 2016

A finite group [Formula: see text] is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup [Formula: see text] with a nontrivial complement [Formula: see text] such that [Formula: see text] for all nonidentity elements [Formula: see text]. Suppose that a finite group [Formula: see text] admits a Frobenius-like group of automorphisms [Formula: see text] of coprime order with [Formula: see text] In case where [Formula: see text] we prove that the groups [Formula: see text] and [Formula: see text] have the same nilpotent length under certain additional assumptions.

Frobenius-like groups as groups of automorphisms

TURKISH JOURNAL OF MATHEMATICS, 2014

A finite group F H is said to be Frobenius-like if it has a nontrivial nilpotent normal subgroup F with a nontrivial complement H such that F H/[F, F ] is a Frobenius group with Frobenius kernel F/[F, F ]. Such subgroups and sections are abundant in any nonnilpotent finite group. We discuss several recent results about the properties of a finite group G admitting a Frobenius-like group of automorphisms F H aiming at restrictions on G in terms of CG(H) and focusing mainly on bounds for the Fitting height and related parameters. Earlier such results were obtained for Frobenius groups of automorphisms; new theorems for Frobenius-like groups are based on new representation-theoretic results. Apart from a brief survey, the paper contains the new theorem on almost nilpotency of a finite group admitting a Frobenius-like group of automorphisms with fixed-point-free almost extraspecial kernel.

Invariant theory of finite general linear groups modulo Frobenius powers

Communications in Algebra, 2018

We prove some cases of a conjecture of Lewis, Reiner and Stanton regarding Hilbert series corresponding to the action of Gl n (F q) on a polynomial ring modulo Frobenius powers. We also give a few conjectures about the invariant ring for certain cases that we don't prove completely.

CHARACTERIZATION OF FROBENIUS GROUPS OF SPECIAL TYPE

2000

We define a Con-Cos group G to be one having a proper normal subgroup N whose cosets other than N itself are conjugacy classes. It follows easily that N = G0, the derived group of G. Most of the paper is devoted to trying to classify finite Con-Cos groups satisfy- ing the additional requirement that N has just two conjugacy

Frobenius Groups with Perfect Order Classes

arXiv (Cornell University), 2021

The purpose of this paper is to investigate the finite Frobenius groups with "perfect order classes"; that is, those for which the number of elements of each order is a divisor of the order of the group. If a finite Frobenius group has perfect order classes then so too does its Frobenius complement, the Frobenius kernel is a homocyclic group of odd prime power order, and the Frobenius complement acts regularly on the elements of prime order in the Frobenius kernel. The converse is also true. Combined with elementary number-theoretic arguments, we use this to provide characterisations of several important classes of Frobenius groups. The insoluble Frobenius groups with perfect order classes are fully characterised. These turn out to be the perfect Frobenius groups whose Frobenius kernel is a homocyclic 11-group of rank 2. We also determine precisely which nilpotent Frobenius complements have perfect order classes, from which it follows that a Frobenius group with nilpotent complement has perfect order classes only if the Frobenius complement is a cyclic {2, 3}-group of even order. Those Frobenius groups for which the Frobenius complement is a biprimary group are also described fully, and we show that no soluble Frobenius group whose Frobenius complement is a {2, 3, 5}group with order divisible by 30 has perfect order classes. Contents

On the characters of ppp-solvable groups

Transactions of the American Mathematical Society, 1961

Introduction. In the theory of group characters, modular representation theory has explained some of the regularities in the behavior of the irreducible characters of a finite group; not unexpectedly, the theory in turn poses new problems of its own. These problems, which are asked by Brauer in [4], seem to lie deep. In this paper we look at the situation for solvable groups. We can answer some of the questions in [4] for these groups, and in doing so, obtain new properties for their characters. Finite solvable groups have recently been the object of much investigation by group theorists, especially with the end of relating the structure of such groups to their Sylow /»-subgroups. Our work does not lie quite in this direction, although we have one result tying up arithmetic properties of the characters to the structure of certain /»-subgroups. Since the prime number p is always fixed, we can actually work in the more general class of /»-solvable groups, and shall do so. Let © be a finite group of order g = pago, where p is a fixed prime number, a is an integer ^0, and ip, go) = 1. In the modular theory, the main results of which are in [2; 3; 5; 9], the characters of the irreducible complex-valued representations of ©, or as we shall say, the irreducible characters of ®, are partitioned into disjoint sets, these sets being the so-called blocks of © for the prime p. Each block B has attached to it a /»-subgroup 35 of © determined up to conjugates in @, the defect group of the block B. If 35 has order pd, in which case we say B has defect d, and if Xn is an irreducible character in B, then the degree of Xk-¡s divisible by p to the exponent a-d+e", where the nonnegative integer e" is defined as the height of Xm-Now let © be a /»-solvable group, that is, © has a composition series such that each factor is either a /»-group or a /»'-group, a /»'-group being one of order prime to p. The following is then true: Let B be a block of © with defect group 3). If 35 is abelian, then every character Xu in B has height 0. Conversely, if B is the block containing the 1-character, and if every character in B has height 0, then 35 is abelian. In particular, this gives a necessary and sufficient condition for the Sylow /»-subgroups of © to be abelian. In the general modular theory of finite groups, each irreducible character X" can be decomposed into a sum of irreducible modular characters <£p of ®,