Structure of the Macdonald groups in one parameter (Q6495780)

In the paper under review, the author consider the so called Macdonald groups in a single parameter (see [\textit{I. D. Macdonald}, Can. J. Math. 14, 602--613 (1962; Zbl 0109.01502)]):NEWLINE\[NEWLINEG(\gamma)=\big \langle A, B \; \big | \; A^{[A,B]}=A^{\gamma}, B^{[B,A]}=B^{\gamma} \big \rangle, \quad \gamma \in \mathbb{Z}, \quad \gamma \not \in \{0,1,2\}.NEWLINE\]NEWLINEThe authors fill a gap in Macdonald's proof that \(G(\gamma)\) is always nilpotent and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of \(G(\gamma)\).NEWLINENEWLINEIn particular (Theorem 3.2) the Macdonald group \(G(\gamma)\) is nilpotent of class at most \(7\) (if \(\gamma \equiv 7 \bmod 9\) and, e.g., \(|G(7)|=2^{4} \cdot 3^{10}\), \(|G(16)|=3^{10}\cdot 5^{7}\), \(|G(25)|=2^{18} \cdot 3^{10}\)). The reviewer remark that, according to the computations presented by the authors, in most cases the groups \(G(\gamma)\) have class \(5\).