The group S₅ is a permutation group on the set {1,2,3,4,5}. But S₅ also permutes pairs from this set. For instance, the permutation (1,2,3) transforms the pair {2,4} into the pair {3,4}.
Let X be the set consisting of all subsets of {1,2,3,4,5} of size 2 (there are 10 such pairs). Because each in S₅ is injective, {(a), (b)} has size 2 whenever {a, b} is a subset of {1,2,3,4,5} of this size. Hence we can define, for each in S₅, the following map:

₂ : X -> X,    {a, b} -> {(a), (b)}.

The map ₂ is bijective: the inverse of ₂ is (^-1)₂. Thus, for an arbitrary element {a, b} from X:

((^-1)₂o ₂ )({a, b}) = (^-1)₂({(a), (b}) = {^-1((a)), ^-1((b))} = {a, b},

and similarly

₂o (^-1)₂({a,b}) = {a,b}.

Finally, we check that the map S₅ -> Sym(X),    -> ₂ is a morphism S₅ -> Sym(X), so that we indeed have a permutation representation. Let , be arbitrary elements of S₅. We need to verify that ( )₂ = ₂₂, that is, that left and right hand side represent the same bijection. This is simple: for each {a,b} in X we have

()₂({a,b}) = {()(a), ()(b)} = {((a)),((b))} = ₂({(a),(b)}) = ₂(₂({a,b})) = (_{2 2})({a,b}).

We name the subsets of {1,2,3,4,5} of size 2 by letters as follows:

a = {1,2}		f = {2,4}
b = {1,3}		g = {2,5}
c = {1,4}		h = {3,4}
d = {1,5}		i = {3,5}
e = {2,3}		j = {4,5}

Thus, for example:

e -> e
(1,2) -> (b,e)(c,f)(d,g),
(1,2,3) -> (a,e,b)(c,f,h)(d,g,i),
(1,2,3,4,5) -> (a,e,h,j,d)(b,f,i,c,g).

Of course we can restrict the action on the pairs of {1,2,3,4,5} to any subgroup of S₅. In particular to A₅.

We generalise the above example as follows. Let X_k be the set of all subsets of {1, ..., n} of size k. Thus,

|Xk| =

n k

.

Each permutation in S_n acts on X_k as follows. The set X in X_k is mapped to the set {(x) | x X}. This defines a bijection _k : X_k -> X_k, and so is an element of Sym(X_k). The map that assigns to S_n the element _k of Sym(X_k), is a morphism S_n -> Sym(X_k) and hence a permutation representation of S_n.

Of course we can restrict the action on X_k to any subgroup of S_n, in particular to A_n.

The general linear group GL_n(R) acts on the set of vectors in Rⁿ. Indeed, if A is in GL_n(R), then A: Rⁿ -> Rⁿ is an invertible map.

Each element of the group D_n of symmetries of the regular n-gon permutes the n vertices of the n-gon. If we number these vertices 1 to n (counter clockwise), the a rotation over 2/n induces the n-cycle

(1,2, ..., n)

on these vertices.

A reflection in the axis through the center of the n-gon and the vertex 1 induces the permutation

(2,n)(3,n - 1) ··· (n/2,n/2 +1)

in case n is even, and

(2,n)(3,n - 1) ··· ((n - 1)/2,(n + 1)/2),

in case n is odd.

This yields a permutation representation of D_n into S_n.