Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Structure of the Macdonald groups in one parameter

Alexander Montoya Ocampo, Fernando Szechtman

Abstract

Consider the Macdonald groups $G(α)=\langle A,B\,|\, A^{[A,B]}=A^α,\, B^{[B,A]}=B^α\rangle$, $α\in{\mathbf Z}$. We fill a gap in Macdonald's proof that $G(α)$ is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of $G(α)$.

Structure of the Macdonald groups in one parameter

Abstract

Consider the Macdonald groups , . We fill a gap in Macdonald's proof that is always nilpotent, and proceed to determine the order, upper and lower central series, nilpotency class, and exponent of .
Paper Structure (11 sections, 25 theorems, 203 equations)

This paper contains 11 sections, 25 theorems, 203 equations.

Table of Contents

  1. Introduction
  2. Valuation calculations
  3. Nilpotence of $G$
  4. More valuation calculations
  5. A presentation of $J$
  6. An upper bound for the order of $J$
  7. Order and basic properties of $J$
  8. Upper and lower central series of $J$
  9. Exponent of $J$
  10. Order, nilpotency class, and exponent of $G(\alpha)$
  11. Appendix

Key Result

Proposition 2.1

Suppose $\alpha>0$, $p\in{\mathbb N}$ is a prime factor of $\alpha-1$, and $v_p(\alpha-1)=m$. Then Moreover, in the last case, we have $\gamma=3^{2+s}t$, where $t\in{\mathbb N}$ and $t\equiv -u\mod 3$.

Theorems & Definitions (49)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 4.1
  • proof
  • ...and 39 more