Generating Sets and a Structure of the Wreath Product of Groups with Non-Faithful Group Action

Abstract

Given a permutational wreath product sequence of cyclic groups, we investigate its minimal generating set, the minimal generating set for its commutator and some properties of its commutator subgroup. We generalize the result presented in the book of J. Meldrum [11] also the results of A. Woryna [4]. The quotient group of the restricted and unrestricted wreath product by its commutator is found. The generic sets of commutator of wreath product were investigated. The structure of wreath product with non-faithful group action is investigated. We strengthen the results from the author [17, 19] and construct the minimal generating set for the wreath product of both finite and infinite cyclic groups, in addition to the direct product of such groups. We generalise the results of Meldrum J. [11] about commutator subgroup of wreath products since, as well as considering regular wreath products, we consider those which are not regular (in the sense that the active group A does not have to act faithfully). The commutator of such a group, its minimal generating set and the center of such products has been investigated here. The minimal generating sets for new class of wreath-cyclic geometrical groups and for the commutator of the wreath product are found.