utc i think daqu is exactly right and if not could somebody please explain why not that is why not say an involution is an element of order a coxeter group is any group generated by finitely many involutions if true that lets somebody know exactly what a coxeter group is without knowing what the word presentation means or worse having to know or accept on faith any theorems about presentations then the stuff about m ij and so on could be presented as consequences of the definition and those who know about presentations could then say ah yes and in fact the class of coxeter groups can be defined via presentations i think in general definitions should involve just as little apparatus as possible and only involve extra machinery when that significantly shortens and simplifies what is being defined an easier definition should never depend on a harder definition of course if i'm wrong and this is not what a coxeter group is somebody please explain and give a counterexample and thanks in advance november