John Cossey

Bio: John Cossey is an academic researcher from Australian National University. The author has contributed to research in topics: Finite group & Permutable prime. The author has an hindex of 16, co-authored 92 publications receiving 715 citations.

Maximal Frattini extensions

Abstract: 1. Motivation. At the recent Santa Cruz conference on finite groups, several participants asked variants of the following question. Given a finite group G, how large a finite group A can be extended by G so tha t A falls into the Fratt ini subgroup of the extension ? (Of course, as Frat t ini subgroups of finite groups are nilpotent, only nilpotent A come into consideration.) A moment ' s thought shows tha t when G has prime order, there are such A of arbitrarily large order, and it is easy to see tha t the same is true in general (provided only that G is not trivial). So the real question is to identify just what measure of the size of A is relevant here. The answer: the cardinalities of the minimal generating sets of the Sylow subgroups, and nothing else. A related question was investigated by Gasehiitz over twenty-five years ago, and this work was inspired by his results.

Classes of finite groups

Abstract: Many group theorists all over the world have been trying in the last twenty-five years to extend and adapt the magnificent methods of the Theory of Finite Soluble Groups to the more ambitious universe of all finite groups. This is a natural progression after the classification of finite simple groups but the achievements in this area are scattered in various papers. Our objectives in this book were to gather, order and examine all this material, including the latest advances made, give a new approach to some classic topics, shed light on some fundamental facts that still remain unpublished and present some new subjects of research in the theory of classes of finite, not necessarily solvable, groups.

Varieties of Groups

Abstract: Roughly speaking, a variety of groups is a class of groups in which certain relations called laws or rules or identities are universally valid. The most widely investigated variety of groups is that of abelian groups in which the commutative law is universally valid or, in other words, in which for arbitrary elements x1, x2 of the group, the identity $$ \left( {{x_1},{x_2}} \right) = 1 $$ holds, where, as usual, (x1, x2) is defined as the commutator $$ x_1^{ - 1}x_2^{ - 1}{x_1}{x_2} $$ of x1 and x2. Similarly, the Burnside variety of exponent e is defined as the class of groups in which, for all elements x, the relation $$ {x^e} = 1 $$ holds. The metabelian groups of Chapter II.6 can be defined as the variety of groups define4 by the identity $$ \left( {\left( {{x_1},{x_2}} \right),\left( {{x_3},{x_4}} \right)} \right) = 1, $$ and the nilpotent groups of class c discussed in Chapter II.7 form the variety defined by the identity $$ \left( {{x_1},{x_2},...,{x_c},{x_{c + 1}}} \right) = 1 $$ where the parentheses ( ) denote the simple (c + l)-fold commutator defined in Chapter II.7.