Sylow subgroups of the Macdonald group on 2 parameters

TL;DR

The paper resolves a gap in Macdonald's original proof by establishing the nilpotence of the Macdonald groups $G(\alpha,\beta)=\langle A,B \mid A^{[A,B]}=A^{\alpha},\ B^{[B,A]}=B^{\beta}\rangle$ and determining the exact order and nilpotency class of each Sylow $p$-subgroup $G(\alpha,\beta)_p$. It achieves this by deriving new relations among $A,B,C=[A,B]$, exploiting a symmetry between $(\alpha,\beta)$ and $(\beta,\alpha)$, and conducting a detailed case analysis guided by the $p$-adic valuations $v_p(\alpha-1)=m$, $v_p(\beta-1)=n$, and $v_p(\alpha-\beta)=\ell$. The main contributions are sharp, case-by-case bounds on $|G_p|$ and the nilpotency class $f$, the identification of the maximal possible class (7) for the Sylow 3-subgroup under $\alpha,\beta\equiv 7\pmod{9}$ and $\alpha\equiv\beta\pmod{27}$, and constructive images realizing these bounds. Collectively, these results show that $G(\alpha,\beta)$ is a direct product of its Sylow subgroups with well-described internal structure, thereby completing Macdonald’s broader program on deficiency-zero 2-generator groups and providing precise information about the finite $p$-group constituents.

Abstract

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

Theorems & Definitions (54)

• Proposition 2.1
• proof
• Lemma 2.2
• proof
• Theorem 3.1
• proof
• Proposition 3.2
• proof
• Corollary 3.3
• Theorem 3.4