On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups (Q2413198)

Summary: 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 \(\mathrm{PG}(3,7)\) and \(\mathrm{PG}(3,8)\).