1
$\begingroup$

Can anyone describe a group with following presentations? (rigorous proof is not needed) $$ \langle x,y,z \mid x^2, y^2, z^2, (xz)^2, (xy)^3, (yz)^3\rangle $$

  • 2
    2012-11-19
  • 0
    Oh, sry please use `$\langle\rangle$` instead of $<>$ ... and use LaTeX.2012-11-19
  • 0
    I would recommend choosing x,y,z to be reflections, in such a way that the products xz, xy, and yz will be rotations with the right order.2012-11-19
  • 2
    See http://en.wikipedia.org/wiki/Coxeter_group .2012-11-19

1 Answers 1

2

Since all generators are involutions, and all other relations are powers of a product of generators, this is a Coxeter group, and it suffices to translate those other relations into a Coxeter diagram. You get a linear diagram with three nodes and simple bonds, which corresponds the the symmetric group $S_4$. Concretely $x,y,z$ will give the adjacent transpositions $(1~2)$, $(2~3)$, and $(3~4)$ respectively.

  • 0
    Yes, the $(2,3,3)$ group I gave is a subgroup of index $2,$ not the whole group.2012-11-19