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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 of a group is a friendly subgroup of if every subgroup of isomorphic to is conjugate in to . We explore easy-to-test sufficient conditions for a subgroup to be a friendly subgroup of a finite group , and for this, present an algorithm for determining whether a finite group is a Sylow tower group. We also classify the maximal partial spreads invariant under a group of order 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. 233–242DOI 10.4171/PM/2004