Anagrammatic quotients of free groups

TL;DR

This paper investigates the structure of a quotient of a free group generated by 26 elements, defined by relations from English language anagrams, revealing a surprisingly simple presentation with specific missing commutators.

Contribution

It provides a detailed description of the quotient group structure and an algorithm to determine such groups from any dictionary, with practical examples from Scrabble.

Findings

The quotient group has a simple presentation with 301 of 325 possible commutators.

All missing commutators involve letters j, q, x, z.

An algorithm is developed to determine the group structure from a dictionary.

Abstract

We determine the structure of the quotient of the free group on 26 generators by English language anagrams. This group admits a surprisingly simple presentation as a quotient of the free group by 301 of the possible 325 commutators of pairs of generators; all of the 24 missing commutators involve at least one of the letters j, q, x, z. We describe the algorithm which can be used to determine this group given any dictionary, and provide examples from the SOWPODS scrabble dictionary witnessing the 301 commutators found.