2: Free Groups and Presentations; Coxeter Groups
- Page ID
- 179991
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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, \(D_{n}\) can be described as the group with generators \(r,s\) and relations
\[r^{n}=e,\quad s^{2}=e,\quad srsr=e. \nonumber \]
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 \(\alpha\colon S\rightarrow S^{\prime}\) of monoids is a map such that \(\alpha(ab)=\alpha(a)\alpha(b)\) for all \(a,b\in S\) and \(\alpha(e)=e\) — unlike the case of groups, the second condition is not automatic. A homomorphism of monoids preserves all finite products.