# Coxeter group generators

Is there a "canonical" set of generators for a given coxeter group? If so, is there a method for going from an arbitrary set of generators of the group to the canonical?

(The "textbook" definition doesn't include this in the definition of these groups, although it certainly seems to use them.)

Thanks.

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The notion of "Coxeter group" incorporates a particular choice of generators and relations for the group. Examples show that the underlying abstract group may well be given by two distinct sets of Coxeter data, which in turn complicates the question of studying isomorphisms and automorphisms. So I'm not sure what the question here actually means. Keep in mind that even a finite symmetric group can usually be given by quite different sets of generators and relations, so a Coxeter presentation of that abstract group is rather special. –  Jim Humphreys Feb 26 '11 at 21:46

My answer is definitely less complete than that of Mark Sapir, but in case you want to see an explicit example: the easiest counterexample is the dihedral group $D_{12}$, which you can view either as a Coxeter group of type $G_2$ (with $2$ generators), or as a Coxeter group of type $A_1 \times A_2$ (with $3$ generators).