Example:
Assume two finite and transitive actions of G on X and Y.
They yield, as was just described, a canonical action of G ´G on
X ´Y which has as
one of its restrictions the action of D(G ´G),
the diagonal, which is isomorphic to G, on X ´Y.
We notice that, for fixed x ÎX, y ÎY, the following is true
(see exercise):
- Each orbit of G on X ´Y contains an element of the
form (x,gy).
- The stabilizer of (x,gy) is Gx ÇgGyg-1, hence the
action of G on the orbit of (x,gy) is similar to the action
of G on G/(Gx ÇgGyg-1) (recall the lemma).
- (x,gy) lies in the orbit
of (x,g'y) if and only if
GxgGy=Gxg'Gy.
Hence the following is true:
Corollary:
If G acts transitively on both X and Y, then, for fixed
x ÎX, y ÎY, the mapping
G \\(X ´Y) -> Gx \G/Gy :G(x,gy) -> GxgGy
is a bijection (note that G \\(X ´Y) stands for
D(G ´G) \\(X ´Y)). Moreover, the action of G on
the orbit G(x,gy) and on the set of left cosets
G/(Gx ÇgGyg-1)
are similar.
Hence, if D denotes a transversal of the set of double cosets Gx \G/Gy,
for fixed x ÎX, y ÎY,
then we have
the following similarity:
G (X ´Y ) » G ( Èg ÎD G/(Gx
ÇgGyg-1) ).
