Sylow subgroups of the Macdonald group on 2 parameters

TL;DR

This paper investigates the structure of Sylow subgroups within the Macdonald group defined by two parameters, clarifying its nilpotency and providing explicit orders and classes of these subgroups.

Contribution

It corrects and completes Macdonald's original proof regarding the nilpotency of the group and determines the order and nilpotency class of its Sylow subgroups.

Findings

Confirmed the nilpotency of G(α,β)

Derived the order of Sylow subgroups

Established the nilpotency class of Sylow subgroups

Abstract

Consider the Macdonald group $G(\alpha,\beta)=\langle A,B\,|\, A^{[A,B]}=A^\alpha,\, B^{[B,A]}=B^\beta\rangle$, where $\alpha$ and $\beta$ are integers different from one. We fill a gap in Macdonald's original proof that $G(\alpha,\beta)$ is nilpotent, and find the order and nilpotency class of each Sylow subgroup of $G(\alpha,\beta)$.