A calculus for a certain class of word problems in groups

Abstract

This paper solves the following problems. Let G be a finitely presented group given by generators g 1, g 2, …, g k and relations R 1=R 2=…=R t=I where R i (i= 1, 2, …, t) are give words in g 1, g 2, …, g k and their inverses. Let H be a subgroup of G of finite index, which is generated by a finite set of words, h 1, h 2, …, h r in the generators of G and their inverses. Find a set of representatives σ 1, σ 2, …, σ 3 of the right cosets of H, and furthermore, given a word W in the generators of G and their inverses, express W in the form W=W *σ i where W * is a word in h 1, h 2, … h r and their inverses. The solution to this problem enables one to solve various interesting problems in group theory. The computation is extremely practical and may be programed easily on a high-speed computing machine. It may be pointed out here that this paper extends the results of [3] in that it gives a systematic procedure for expressing the Schreier generators of a subgroup as words in the originally given generators. The process was used in an informal way as far back as 1960, but at that time machine computation was not envisaged.