arXiv subjects
Publications and source records attributed to Paul Flavell.
Glauberman's characteristic $p$-functor $K^{\infty}$ has a number of remarkable properties. For example it controls $p$-transfer in any finite group provided $p \geq 5$. These notes are the author's attempt to understand and simplify this work.
We present a new proof of the theorem of Dickson that gives generators for the groups $SL_{2}(q)$ where $q$ is an odd prime power.
We give a self contained and concise proof of the following result, which was first proved by Glauberman as his $ZJ$-Theorem. Let $p$ be an odd prime and $S \not= 1$ a $p$-group. Then there exists a characteristic subgroup $W(S) \not= 1$ of $S$ with the property: whenever $G$ is a finite $Qd(p)$-free group of characteristic $p$ and $S$ is a Sylow subgroup of $G$ then $W(S) \unlhd G$.
We give another proof of the classic result of Dickson that determines the finite subgroups of 2-dimensional projective general linear groups. In fact we go a little further and give an abstract characterization of those subgroups in terms of permutation groups.
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 McBride's Nonsolvable Signalizer Functor Theorem. Given their fundamental role, it is desirable to have new and different proofs of them. This is accomplished in {\em A new proof of the Solvable Signalizer Functor Theorem,} P. Flavell, J. Algebra, 398 (2014) 350--363 for Glauberman's Theorem. The purpose of this paper is to give a new proof of McBride's Theorem.
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 article, the $p$-solvable restriction will be weakened to a $K$-group hypothesis. An application of this work will be a new proof of the Nonsolvable Signalizer Functor Theorem.
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 $G$ and the collection of fixed point subgroups $\{ C_{G}(a) \;|\; a \in A^{\#} \}$. An application of this result will be a new proof of McBride's Nonsolvable Signalizer Functor Theorem.
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, 'Near solvable signalizer functors on finite groups' J. Algebra {\bf 78}(1) (1982) 181-214 and 'Nonsolvable signalizer functors on finite groups', J. Algebra {\bf 78}(1) (1982) 215-238 are extended. One purpose of this work is to build a general theory of automorphisms, one of whose applications will be a new proof of the Nonsolvable Signalizer Functor Theorem. As an illustration, this work concludes with a new proof of a special case of that theorem due to Gorenstein and Lyons.
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.
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 that do not use the Classification theorem. The most direct proof gives a value of $k=10$. By lengthening one of our arguments slightly, we obtain a value of $k=7$.