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. 
 A: 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).
A: There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups.
Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.  
