In most undergraduate courses, groups are first introduced as a primarily *algebraic* concept – a set equipped with a number of algebraic operations (group multiplication, multiplicative inverse, and multiplicative identity) and obeying a number of rules of algebra (most notably the associative law). It is only somewhat later that one learns that groups are not solely an algebraic object, but can also be equipped with the structure of a manifold (giving rise to *Lie groups*) or a topological space (giving rise to *topological groups*). (See also this post for a number of other ways to think about groups.)

Another important way to enrich the structure of a group $latex {G}&fg=000000$ is to give it some *geometry*. A fundamental way to provide such a geometric structure is to specify a list of generators $latex {S}&fg=000000$ of the group $latex {G}&fg=000000$. Let us call such a pair $latex {(G,S)}&fg=000000$…