Classification of Group Extensions

Abstract

This PhD thesis studies the classification of group extensions, whose main result is the classification, up to isomorphism, of finite nonsolvable groups of order less than 23,040. To make this classification computationally effective, we develop reduction methods that replace extension computations involving a finite nonsolvable group by computations in suitable smaller solvable subgroups. This reduction from nonsolvable to solvable computations led to a major decrease in computing time and made it possible, on a standard personal computer available in 2002, to classify around 8.4 million nonsolvable groups of order less than 23,040. A nonsolvable group E is constructed from a given perfect kernel P, its perfect residual, extended by a given solvable quotient H=E/P. The isomorphism problem for the resulting nonsolvable groups E is reduced to isomorphism computations between extensions involving a small nilpotent subgroup U<P and the quotient H. We introduce and use (Zϕ)-extensions and supplement methods to obtain explicit classifications in a range of finite cases. These methods are applied to the classification of finite nonsolvable groups of order less than 23,040 and provide algorithms for constructing and identifying extensions up to isomorphism.