### Products of Actions

Now we take *two* actions
into account, _{G}X and _{H}Y, say, and derive further actions from these.
Without loss of generality we can assume *X ÇY= Æ* since
otherwise we can rename the elements of *X*, in order to replace _{G}X by a
similar action _{G}X', for which *X' ÇY= Æ*.
Now we form the
(disjoint) union *X È Y* and let *G ´H* act on this set as follows:

*
(G ´H) ´(X È Y) -> X È Y
*

where *((g,h),z) -> gz* if *z ÎX* and *((g,h),z) -> hz* if
*z ÎY*.
The corresponding permutation group will be denoted
by * bar (G) Åbar (H)* (cf. the formula) and called the *direct sum*
of * bar (G)* and * bar (H)*.
Another canonical action of *G ´H* is that on the cartesian product:

*
(G ´H) ´(X ´Y) -> X ´Y :((g,h),(x,y)) -> (gx,hy).
*

The corresponding permutation group will be denoted
by * bar (G) Äbar (H)* and called the *cartesian product*
of * bar (G)* and * bar (H)*.
An important particular case is

**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 *G*_{x} ÇgG_{y}g^{-1}, hence the
action of *G* on the orbit of *(x,gy)* is similar to the action
of *G* on *G/(G*_{x} ÇgG_{y}g^{-1}) (recall the lemma).
*(x,gy)* lies in the orbit
of *(x,g'y)* if and only if
*G*_{x}gG_{y}=G_{x}g'G_{y}.

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) -> G*_{x} \G/G_{y} :G(x,gy) -> G_{x}gG_{y}

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/(G*_{x} ÇgG_{y}g^{-1})

are similar.
Hence, if * D* denotes a transversal of the set of double cosets *G*_{x} \G/G_{y},
for fixed *x ÎX, y ÎY*,
then we have
the following similarity:
_{G} (X ´Y ) » _{G} ( È_{g ÎD} G/(G_{x}
ÇgG_{y}g^{-1}) ).

last changed: January 19, 2005