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The automorphism group of finite $2$-groups associated to the Macdonald group

Alexander Montoya Ocampo, Fernando Szechtman

Abstract

We consider the Macdonald group $\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}\rangle$ and its Sylow 2-subgroup $J=\langle x,y\,|\, x^{[x,y]}=x^{1+2^m\ell},\, y^{[y,x]}=y^{1+2^m\ell}, x^{2^{3m-1}}=y^{2^{3m-1}}=1\rangle$, where $m\geq 1$ and $\ell$ is odd. Then $J$ has order $2^{7m-3}$, and nilpotency class 5 if $m>1$ and 3 if $m=1$. We determine the automorphism group of the 2-groups $J$, $H=J/Z(J)$ and $K=H/Z(H)$, where $|H|=2^{6m-3}$ and $|K|=2^{5m-3}$. Explicit multiplication, power, and commutator formulas for $J$, $H$, and $K$ are given, and used in the calculation of $\mathrm{Aut}(J)$, $\mathrm{Aut}(H)$, and $\mathrm{Aut}(K)$.

The automorphism group of finite $2$-groups associated to the Macdonald group

Abstract

We consider the Macdonald group and its Sylow 2-subgroup , where and is odd. Then has order , and nilpotency class 5 if and 3 if . We determine the automorphism group of the 2-groups , and , where and . Explicit multiplication, power, and commutator formulas for , , and are given, and used in the calculation of , , and .
Paper Structure (8 sections, 42 theorems, 232 equations)

This paper contains 8 sections, 42 theorems, 232 equations.

Table of Contents

  1. Introduction
  2. Background on the Sylow 2-subgroup of the Macdonald group
  3. Structural formulas for $J$
  4. Order of the terms of the upper central series of $J$
  5. The automorphism group of $J/Z_2(J)$
  6. The automorphism group of $J/Z_1(J)$
  7. The automorphism group of $J$
  8. Appendix

Key Result

Theorem 3.1

For all $n,t\in{\mathbb Z}$ the following identities hold in $J$: where $\textup{exp} A = -2s\ell\phi(n)t$, $\textup{exp} B = 2s\ell n\phi(t)$, $\textup{exp}\,C = nt - 2s\ell\phi(n)\phi(t)$ and

Theorems & Definitions (73)

  • Theorem 3.1
  • Theorem 3.2
  • Corollary 3.3
  • Proposition 3.4
  • Theorem 3.5
  • Proposition 5.1
  • proof
  • Proposition 5.2
  • proof
  • Corollary 5.3
  • ...and 63 more