Products of Actions |

where(G ´H) ´(X È Y) -> X È Y

The corresponding permutation group will be denoted by(G ´H) ´(X ´Y) -> X ´Y :((g,h),(x,y)) -> (gx,hy).

Example:Assume two finite and transitive actions ofGonXandY. They yield, as was just described, a canonical action ofG ´GonX ´Ywhich has as one of its restrictions the action ofD(G ´G), the diagonal, which is isomorphic toG, onX ´Y. We notice that, for fixedx ÎX, y ÎY, the following is true (see exercise):

- Each orbit of
GonX ´Ycontains an element of the form(x,gy).- The stabilizer of
(x,gy)isG, hence the action of_{x}ÇgG_{y}g^{-1}Gon the orbit of(x,gy)is similar to the action ofGonG/(G(recall the lemma)._{x}ÇgG_{y}g^{-1})(x,gy)lies in the orbit of(x,g'y)if and only ifG_{x}gG_{y}=G_{x}g'G_{y}.Hence the following is true:

Corollary:IfGacts transitively on bothXandY, then, for fixedx ÎX, y ÎY, the mappingis a bijection (note thatG \\(X ´Y) -> G_{x}\G/G_{y}:G(x,gy) -> G_{x}gG_{y}G \\(X ´Y)stands forD(G ´G) \\(X ´Y)). Moreover, the action ofGon the orbitG(x,gy)and on the set of left cosetsare similar. Hence, ifG/(G_{x}ÇgG_{y}g^{-1})Ddenotes a transversal of the set of double cosetsGfor fixed_{x}\G/G_{y},x ÎX, y ÎY, then we have the following similarity:_{G}(X ´Y ) »_{G}( È_{g ÎD}G/(G_{x}ÇgG_{y}g^{-1}) ).

harald.fripertinger@kfunigraz.ac.at,

last changed: August 28, 2001

Products of Actions |