Structure of the Macdonald groups in one parameter

TL;DR

This paper completes Macdonald's proof that certain groups are nilpotent and provides detailed structural properties such as order, series, class, and exponent for these groups.

Contribution

It fills a gap in the proof of nilpotency and thoroughly determines key structural characteristics of Macdonald groups.

Findings

Confirmed nilpotency of Macdonald groups.

Calculated order, central series, and nilpotency class.

Determined the exponent of the groups.

Abstract

Consider the Macdonald groups $G(\alpha)=\langle A,B\,|\, A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\alpha\rangle$, $\alpha\in{\mathbf Z}$. We fill a gap in Macdonald's proof that $G(\alpha)$ is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of $G(\alpha)$.