2: Free Groups and Presentations; Coxeter Groups
( \newcommand{\kernel}{\mathrm{null}\,}\)
It is frequently useful to describe a group by giving a set of generators for the group and a set of relations for the generators from which every other relation in the group can be deduced. For example, Dn can be described as the group with generators r,s and relations
rn=e,s2=e,srsr=e.
In this chapter, we make precise what this means. First we need to define the free group on a set X of generators — this is a group generated by X and with no relations except for those implied by the group axioms. Because inverses cause problems, we first do this for monoids. Recall that a monoid is a set S with an associative binary operation having an identity element e. A homomorphism α:S→S′ of monoids is a map such that α(ab)=α(a)α(b) for all a,b∈S and α(e)=e — unlike the case of groups, the second condition is not automatic. A homomorphism of monoids preserves all finite products.