Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Proving the 5-Engel identity in the 2-generator group of exponent four

Colin Ramsay

Abstract

It is known that the fifth Engel word $E_5$ is trivial in the 2-generator group of exponent four $B(2,4)$, and so can be written as a product of fourth powers. Explicit products of 250 and 28 powers are known, using fourth powers of words up to lengths four and ten respectively. Using a reduction technique based on the recursive enumerability of the set of trivial words in a finite presentation we were able to rewrite $E_5$ as a product of 26 fourth powers of words up to length five.

Proving the 5-Engel identity in the 2-generator group of exponent four

Abstract

It is known that the fifth Engel word is trivial in the 2-generator group of exponent four , and so can be written as a product of fourth powers. Explicit products of 250 and 28 powers are known, using fourth powers of words up to lengths four and ten respectively. Using a reduction technique based on the recursive enumerability of the set of trivial words in a finite presentation we were able to rewrite as a product of 26 fourth powers of words up to length five.
Paper Structure (3 sections, 1 theorem, 4 equations, 2 tables)

This paper contains 3 sections, 1 theorem, 4 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Generating Proofs
  3. Results

Key Result

Theorem 1

Any reduced proof word $U$ can be generated from $\varepsilon$ by a series of relator append and conjugation moves.

Theorems & Definitions (2)

  • Theorem 1
  • proof