Generation of wreath products and their generalisation

Abstract

This thesis studies generation of iterated wreath products of permutation groups and their generalisations. We first investigate the minimum number of generators of iterated wreath products constructed from the natural action of cyclic groups, alternating groups and symmetric groups. Then we turn to the iterated wreath products of almost simple groups with respect to their regular action and determine the minimum number of generators of such products. We also analyse the structure of generalised wreath products with respect to a sequence of permutation groups indexed by a partially ordered set, which were first introduced by Bailey et al. If the generalised wreath product is constructed from a sequence of symmetric groups with natural action, and the sequence is indexed by a finite partially ordered set, then the minimum number of generators of such a generalised wreath product is the cardinality of the poset. In the final part of this thesis, we will introduce iterated wreath products of transformation semigroups. For iterated wreath products of full transformation semigroups with respect to their natural action, we determine that the rank of such a iterated wreath product is equal to the number of factors that the product involves.

Date of Award	2 Jul 2026
Original language	English
Awarding Institution	University of St Andrews
Supervisor	Martyn Quick (Supervisor)