Structure of the Macdonald groups in one parameter

Abstract Consider the Macdonald groups G ⁢ ( α ) = ⟨ A , B ∣ A [ A , B ] = A α , B [ B , A ] = B α ⟩ G(\alpha)=\langle A,B\mid A^{[A,B]}=A^{\alpha},\,B^{[B,A]}=B^{\alpha}\rangle , α ∈ Z \alpha\in\mathbb{Z} . We fill a gap in Macdonald’s proof that G ⁢ ( α ) G(\alpha) is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of G ⁢ ( α ) G(\alpha) .