The Pseudocentre of a Group (with an appendix by Anthony Genevois)

TL;DR

This paper explores the complex structure of the pseudocentre in various groups, revealing its unpredictable behavior and its deep connections with group properties, extending Wiegold's initial finite group results to broader classes.

Contribution

The work demonstrates that the pseudocentre's structure varies widely across different groups, providing new insights and examples that challenge previous assumptions about its relation to other group invariants.

Findings

A soluble group equals its pseudocentre iff it is abelian.

The pseudocentre of unitriangular matrices is the largest abelian upper central series term.

Wreath products can have an arbitrarily complex pseudocentre.

Abstract

In 1973, Jim Wiegold introduced the concept of pseudocentre P(G) of a group G as the intersection of the normal closures of the centralizers of its elements. He proved that the pseudocentre of a non-trivial finite group is always non-trivial, giving a new variable on which one can use induction in finite group theory. In the same paper, Wiegold states that no obvious relations seem to hold between the pseudocentre and the canonical characteristic subgroups of a group. The aim of this work is to show that the pseudocentre is indeed much more involved in the structure of an arbitrary group then anyone could have expected. For example, we prove that a soluble group coincides with its pseudocentre if and only if it is abelian, and that the structure of the commutator subgroup strongly influences the structure of the pseudocentre. And this is not the end of the story. In fact, the behaviour of the pseudocentre in arbitrary (possibly infinite) groups can be extremely wild: sometimes it is very difficult even to understand whether the pseudocentre is trivial or not. This wilderness is exampled by some of our main results (see the introduction for a complete list): 1) There exists a polycyclic group of Hirsch length 3 in which the pseudocentre is trivial. 2) The pseudocentre of the group of unitriangular matrices over any field is the largest term of the upper central series that is abelian. 3) Free products have a trivial pseudocentre, but there exist amalgamated free products of non-trivial groups coinciding with their pseudocentre. 4) Weakly regular branch groups have a trivial pseudocentre. 5) The pseudocentre of the Thompson group is the derived subgroup. 6) Wreath products can have a totally arbitrary pseudocentre.