A note on groups in which the centraliser of every element of order 5 is a 5-group

TL;DR

This paper investigates the structure of certain groups where elements of order 5 have centralizers that are 5-groups, proving nilpotency of class at most two under specific conditions and correcting previous misconceptions.

Contribution

It establishes that odd order groups with a fixed point free action of A5 of order 5 are nilpotent of class at most two, correcting earlier results and providing detailed group structure insights.

Findings

Groups of odd order with fixed point free A5 action are nilpotent of class ≤2.

Existence of class two r-groups for primes r ≠ 5 admitting A5 action.

Correction of previous claims about the non-existence of certain class two groups.

Abstract

The main theorem in this article shows that a group of odd order which admits the alternating group of degree 5 with an element of order 5 acting fixed point freely is nilpotent of class at most two. For all odd primes r, other than 5, we give a class two r-group which admits the alternating group of degree 5 in such a way. This theorem corrects an earlier result which asserts that such class two groups do not exist. The result allows us to state a theorem giving precise information about groups in which the centralizer of every element of order 5 has centralizer a 5-group.