Definition 3

(W, S) is said to be a Coxeter system if it satisfies the following condition:

(C) For s, s' in S, let m(s, s') be the order of ss' and let I be the set of pairs (s, s') such that m(s, s') is finite. The generating set S and the relations (ss')^{m(s,s')} = 1 for (s, s') in I form a presentation¹ of the group W.

When (W, S) is a Coxeter system, one also says, by abuse of language, that W is a Coxeter group.