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A study of the Wamsley group and its Sylow subgroups

A. Previtali, F. Szechtman

TL;DR

This work analyzes the finite Wamsley group $W$ given by a balanced 3-generator, 3-relator presentation, relating it to the Macdonald group $G$ via a surjective homomorphism and kernel $N$. By reducing to Sylow subgroups $W_p$ through the decomposition $G=\prod_p G_p$ and studying the corresponding $N_p$, the authors obtain explicit presentations, normal forms, and nilpotency data for each $W_p$ in the crucial case $\alpha=\beta$, then assemble these to determine $|W|$ and the derived length of $W$. They deploy two complementary approaches to commutator calculations: direct lower-central-series analysis with Witt’s formula and an algorithmic Cant–Eick/Hall–polynomial framework, producing closed formulas for $[a^i,b^j]$ and explicit normal closures $N_p$ in the various primes. The results culminate in a complete order formula for $W$, detailed nilpotency-class data for its Sylow subgroups, and a crisp determination of the global derived length, advancing the structural understanding of Wamsley-type balanced presentations and their Sylow structure.

Abstract

We study the Wamsley group $\langle X,Y,Z\,|\, X^Z=X^α, {}^Z Y=Y^β, Z^γ=[X,Y]\rangle$ and its Sylow subgroups, where $α^γ\neq 1\neq β^γ$ and $γ>0$, obtaining the sharpest results when $α=β$.

A study of the Wamsley group and its Sylow subgroups

TL;DR

This work analyzes the finite Wamsley group given by a balanced 3-generator, 3-relator presentation, relating it to the Macdonald group via a surjective homomorphism and kernel . By reducing to Sylow subgroups through the decomposition and studying the corresponding , the authors obtain explicit presentations, normal forms, and nilpotency data for each in the crucial case , then assemble these to determine and the derived length of . They deploy two complementary approaches to commutator calculations: direct lower-central-series analysis with Witt’s formula and an algorithmic Cant–Eick/Hall–polynomial framework, producing closed formulas for and explicit normal closures in the various primes. The results culminate in a complete order formula for , detailed nilpotency-class data for its Sylow subgroups, and a crisp determination of the global derived length, advancing the structural understanding of Wamsley-type balanced presentations and their Sylow structure.

Abstract

We study the Wamsley group and its Sylow subgroups, where and , obtaining the sharpest results when .
Paper Structure (12 sections, 40 theorems, 270 equations)

This paper contains 12 sections, 40 theorems, 270 equations.

Table of Contents

  1. Introduction
  2. Approaching $W$ via $G$
  3. The case when $\alpha=\beta$
  4. Case 1 when $\alpha=\beta$
  5. Witt's Formula
  6. An altentative proof of the commutator formula
  7. Case 2 when $\alpha=\beta$
  8. Commutator calculations in Case 3
  9. Case 3 when $\alpha=\beta$
  10. The order of $W$
  11. The nilpotency class of $W_p$
  12. Derived length of $W$

Key Result

Theorem 2.1

Let $T$ be an arbitrary group and $L$ a cyclic group of finite order $n\in{\mathbb N}$. Suppose that $t\in T$ and that $\Lambda$ is an automorphism of $T$ fixing $t$ and such that $\Lambda^n$ is conjugation by $t$. Then there is a group $E$ containing $T$ as a normal subgroup, such that $E/T\cong L$

Theorems & Definitions (69)

  • Theorem 2.1
  • Theorem 2.2
  • proof
  • Theorem 2.3
  • Theorem 2.4
  • proof
  • Proposition 3.1
  • proof
  • Theorem 4.1
  • Proposition 4.2
  • ...and 59 more