Examples
The general linear group and special linear group acting on vectors.
A subgroup of Sym(X) acting on all subsets of X.
The group G = GL(Rn) of real
invertible n×n-matrices acts as a permutation
group on the set of vectors of Rn: the
element g of G is mapped to the bijection
g: Rn -> Rn,
v -> gv , an element of Sym(Rn) .
See a previous example.
As the zero vector is fixed by all matrices in G, the group G is also a permutation group on the set of nonzero vectors.
Also the special linear group SL(Rn), which consists of the n×n-matrices with determinant 1, acts on the nonzero vectors in Rn.
Suppose G acts on X. Then there is an action of G on
the set of all subsets of X. If
Y={x1, ... , xk} is a subset of
X of size k, its image under the permutation g
is
g(Y) =
{g(x1), ... , g(xk)}.
Verify that this defines a permutation representation indeed!
Let Z be the set of all subsets of
X of size 2.
Then Z is G-invariant, that is, for each g in G,
the image g(Y) of a 2-set Y is again a 2-set.
Thus, we find a permutation representation of G on Z.