Now we take two actions
into account, 
and 
, say, and derive further actions from these.
Without loss of generality we can assume 
since
otherwise we can rename the elements of 
, in order to replace by a
similar action 
, for which 
.
Now we form the
(disjoint) union 
and let 
act on this set as follows:
The corresponding permutation group will be denoted
by 
(cf. 
) and called the direct sum
of 
and 
.
Another canonical action of is that on the cartesian product:
The corresponding permutation group will be denoted
by 
and called the cartesian product
of and .
An important particular case is
.
Example
Assume two finite and transitive actions of 
on and 
.
They yield, as was just described, a canonical action of 
on
which has as
one of its restrictions the action of 
,
the diagonal, which is isomorphic to , on .
We notice that, for fixed 
, the following is true
(exercise ):
on contains an element of the
form 
.
is 
, hence the
action of on the orbit of is similar to the action
of on 
(recall ).
lies in the orbit
of 
if and only if
Hence the following is true:
is a bijection (note that
are similar.
Hence, if
.
Corollary
If acts transitively on both and , then, for fixed
, the mapping
stands for
). Moreover, the action of on
the orbit 
and on the set of left cosets
denotes a transversal of the set of double cosets 
for fixed ,
then we have
the following similarity:
Exercises
.