Conjugacy classes and centralisers in wreath products

TL;DR

This paper generalizes the description of conjugacy classes and centralisers in wreath products, extending previous results to cases with infinite base groups and providing efficient algorithms for finite cases.

Contribution

It introduces a generalized framework for conjugacy classes and centralisers in wreath products with infinite base groups and finite top groups, along with algorithms for finite cases.

Findings

Explicit parameterization of conjugacy classes and centralisers.

Efficient algorithms for conjugacy and centraliser computations in finite wreath products.

Extension of classical results to infinite base groups.

Abstract

In analogy to the disjoint cycle decomposition in permutation groups, Ore and Specht define a decomposition of elements of the full monomial group and exploit this to describe conjugacy classes and centralisers of elements in the full monomial group. We generalise their results to wreath products whose base group need not be finite and whose top group acts faithfully on a finite set. We parameterise conjugacy classes and centralisers of elements in such wreath products explicitly. For finite wreath products, our approach yields efficient algorithms for finding conjugating elements, conjugacy classes, and centralisers.