Bounds for the relative n-th nilpotency degree in compact groups

TL;DR

This paper extends bounds on the probability that two elements commute in compact groups, generalizing classical and recent results to better understand the structure of such groups.

Contribution

It improves existing bounds for the relative n-th nilpotency degree specifically within the setting of compact groups.

Findings

Established new bounds for the relative n-th nilpotency degree in compact groups.

Generalized previous results from finite groups to compact groups.

Enhanced understanding of the probabilistic structure of compact groups.

Abstract

The line of investigation of the present paper goes back to a classical work of W. H. Gustafson of the 1973, in which it is described the probability that two randomly chosen group elements commute. In the same work, he gave some bounds for this kind of probability, providing information on the group structure. We have recently obtained some generalizations of his results for finite groups. Here we improve them in the context of the compact groups.