### Action on k-subsets

Further actions of *G* which can be derived from _{G}X are the
actions of *G* on the sets

*[X choose k]:= {M ÍX | | M | =
k },
*

of *k*- *subsets*
of *X*, *1 <= k <= | X | *, which are
defined as follows:

*
G ´[X choose k] -> [X choose k] :(g,M) -> bar (g)M=
{gm | m ÎM },
*

The action _{G}X is called *k*- *homogeneous* if and only
if the corresponding
action of *G* on *[X choose k]* is transitive. An obvious example is the natural
action of *S*_{X} on *X*, it is *k*-homogeneous for *k <= | X | *.

last changed: January 19, 2005