From a given action we can
derive various other actions in a natural
way, e.g. 
yields 
, 
being the
homomorphic image of 
in 
, which was already mentioned. We also
obtain the subactions
on subsets 
which
are nonempty unions of orbits. Furthermore there are the restrictions
to the subgroups 
of . As the orbits of are unions
of orbits of , the comparison of actions and restrictions is a
suitable way of generalizing or specializing structures if they can be
defined as orbits. The following example will show what is meant by this.
The orbits 
.
Example
Let denote a subgroup of the direct product
. Then U acts on as follows:
of this action are called
the bilateral classes
of with respect to . By specializing
we obtain various interesting group theoretical structures some of
which have been mentioned already:
is a subgroup of , then both 
and 
are subgroups of . Their orbits are the subsets
the right cosets
of in , and
the left cosets of in .
denotes a second subgroup
of , then we can put equal to the subgroup 
,
obtaining as orbits
the
- double cosets
of :
is its diagonal
subgroup
Its orbits are the conjugacy classes:
Hence left and right cosets, double cosets and conjugacy classes turn out to be special cases of bilateral classes. Being orbits, two of them are either equal or disjoint, moreover, their order is the index of the stabilizer of an element. We have mentioned this in connection with conjugacy classes and centralizers of elements, here is the consequence for double cosets: Since
we obtain