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

### Leonard H. Soicher

Queen Mary University of London, UK

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## 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).

## Cite this article

Leonard H. Soicher, On classifying objects with specified groups of automorphisms, friendly subgroups, and Sylow tower groups. Port. Math. 74 (2017), no. 3, pp. 233–242

DOI 10.4171/PM/2004