Sylow subgroups of the Macdonald group on 2 parameters

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 $G(\alpha,\beta)$.