On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups

Abstract

We describe some group theory which is useful in the classification of combinatorial objects having given groups of automorphisms. In particular, we show the usefulness of the concept of a friendly subgroup: a subgroup $H$ of a group $K$ is a friendly subgroup of $K$ if every subgroup of $K$ isomorphic to $H$ is conjugate in $K$ to $H$. We explore easy-to-test sufficient conditions for a subgroup $H$ to be a friendly subgroup of a finite group $K$, and for this, present an algorithm for determining whether a finite group $H$ is a Sylow tower group. We also classify the maximal partial spreads invariant under a group of order $5$ in both PG(3,7) and PG (3,8).