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

Ruslan Skuratovskii

doi:10.28924/2291-8639-18-2020-104

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.

Highlights

The form of commutator presentation [11] has been given here in the form of wreath recursion [10] and its commutator width has been studied
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 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

Summary

Introduction

The form of commutator presentation [11] has been given here in the form of wreath recursion [10] and its commutator width has been studied. The results about commutators’s structure given in [11] were improved. [6] tell us that the wreath product Cpn−1 G is n-generated. One of the goal of our research is to study the center and commutator subgroup of wreath product with non-faithful action of active group on the set. As the goal of our paper is the minimal generating set and upper bound of minimal size of the generating set of the commutator subgroup of such class of group. The structure of center and quotient group by its commutator subgroup for a of such non-regular wreath product were still not investigated

Prelimenaries

Minimal generating set of direct product of wreath products of cyclic groups

Conclusion