Paul Flavell

• University of Birmingham

An important result with many applications in the theory of finite groups is the following: Let S ≠ 1 S \not =1 be a finite p-group for some prime p. Then S S contains a characteristic subgroup W ( S ) ≠ 1 W(S) \not = 1 with the property that W ( S ) W(S) is normal in every finite group G G of characteristic p p with S ∈ S y l p ( G ) S \in Syl_p(G...

Let $A$ be an elementary abelian $r$-group with rank at least $3$ that acts faithfully on the finite $r'$-group $G$. Assume that $G$ is $A$-simple, so that $G = K_{1} \times\cdots\times K_{n}$ where $K_{1},\ldots,K_{n}$ is a collection of simple subgroups of $G$ that is permuted transitively by $A$. The purpose of this paper is to characterize $G$...

In 'Primitive pairs of $p$-solvable groups', J. Algebra 324 (2010) 841-859, the author proved a non existence theorem for certain types of amalgams of $p$-solvable groups in the presence of operator groups acting coprimely on the groups in the amalgam. An application of that work was a new proof of the Solvable Signalizer Functor Theorem. In this a...

This work is a continuation of Automorphisms of $K$-groups I, P. Flavell, preprint. The main object of study is a finite $K$-group $G$ that admits an elementary abelian group $A$ acting coprimely. For certain group theoretic properties $\mathcal P$, we study the $AC_{G}(A)$-invariant $\mathcal P$-subgroups of $G$. A number of results of McBride, 'N...

This is the first in a sequence of papers that will develop the theory of automorphisms of nonsolvable finite groups. The sequence will culminate in a new proof of McBride's Nonsolvable Signalizer Functor Theorem, which is one of the fundamental results required for the proof of the Classification of the Finite Simple Groups.

Let $R$ be a group of prime order $r$ that acts on the $r'$-group $G$, let $RG$ be the semidirect product of $G$ with $R$, let ${\mathbb {F}}$ be a field and $V$ be a faithful completely reducible $\mathbb {F}[{RG}]$-module. Trivially, $C_{G}({R})$ acts on $C_{V}({R})$. Let $K$ be the kernel of this action. What can be said about $K$? This question...

We show that there is no absolute bound on the Fitting height of a group with two Sylow numbers.

Let R be a cyclic group of prime order which acts on the extraspecial group F in such a way that F=[F,R]F=[F,R]. Suppose RF acts on a group G so that CG(F)=1CG(F)=1 and (|R|,|G|)=1(|R|,|G|)=1. It is proved that F(CG(R))⊆F(G)F(CG(R))⊆F(G). As corollaries to this, it is shown that the Fitting series of CG(R)CG(R) coincides with the intersections of C...

An odd nilpotent injector of a finite group G is defined to be a subgroup which is maximal subject to being nilpotent of odd order and containing a subgroup of maximal order amongst all abelian subgroups of odd order. We prove that the odd nilpotent injectors of a minimal simple group are all conjugate, extending the result from soluble groups. The...

Let G be a finite soluble group and P a subgroup of order 3. In this article we prove some results about the soluble groups generated by 2 conjugates of P and we use these results to produce some properties of G.

The subgroup structure of a finite group G can be quite complex. However, in the presence of a group R of automorphisms of G and provided we consider only certain subgroups related to R, the situation appears much simpler. The purpose of this article is to provide concrete results in this direction. These results have been applied to give a new pro...

Let r be a prime and suppose the r-group R acts as a group of automorphisms on the r′r′-group G. We study the RCG(R)RCG(R)-invariant subgroups of G and how they influence the structure of G. An application to the study of the automorphism group of a simple group is presented.

The classic Zipper lemma of Wielandt on subnormal subgroups is generalized. An application is a new proof of a theorem of Bartels which describes the subnormal closure of a subgroup.

We prove that there exists a constant $k$ with the property: if $\calC$ is a conjugacy class of a finite group $G$ such that every $k$ elements of $\calC$\ generate a solvable subgroup then $\calC$ generates a solvable subgroup. In particular, using the Classification of Finite Simple Groups, we show that we can take $k=4$. We also present proofs t...

Let G be a fnite group and P a subgroup of order 3. In this paper we proved some results about the soluble subgroups generated by three conjugates of P and we use these results to produce some properties of the group G.

Let G be a finite soluble group that is generated by a conjugacy class consisting of elements of order 3. We show that there exist four conjugates of an element of order 3 that generate a subgroup with the same Fitting height as G. We use this result to find a soluble analogue of the Baer-Suzuki theorem in the case prime 3.

A group G is called an ATI-group if for any abelian subgroup A of G, A∩Ax=1 or A for all x∈G. In this paper the finite ATI-groups are classified.

The Signalizer Functor Method as developed by Gorenstein and Walter played a fundamental role in the first proof of the Classification of the Finite Simple Groups. It plays a similar role in the new proof of the Classification in the Gorenstein-Lyons-Solomon book series. The key results are Glauberman's Solvable Signalizer Functor Theorem and McBri...

We study how the fixed point subgroup of an automorphism influences the structure of a group.

We shall extend a fixed point theorem of Shult to arbitrary finite groups. This will have applications to the study of group automorphisms.

We prove that a finite group G is \( \cal N \) -constrained if and only if it contains a nilpotent subgroup I satisfying \( C_{G}(I \cap I^{g}) \leq I \cap I^{g} \) for all \( g \in G \) .

Let P be an odd $\pi $-group that acts as a group of automorphisms on the soluble $\pi '$-group G. We obtain generators for the fixed points of P on [G, P].

If p is a prime, then a finite group is p-soluble if each of its composition factors is either a p-group or has order coprime to p. For example, soluble groups are p-soluble. However, there are many insoluble groups that are p-soluble. We shall prove the following result. 1991 Mathematics Subject Classification 20D10.

Thesis (Ph. D.)--University of Oxford, 1990.