Permutation representation

In mathematics, the term permutation representation of a (typically finite) group formula_1 can refer to either of two closely related notions: a representation of formula_1 as a group of permutations, or as a group of permutation matrices. The term also refers to the combination of the two.
Abstract permutation representation.
A permutation representation of a group formula_1 on a set formula_4 is a homomorphism from formula_1 to the symmetric group of formula_4:
The image formula_8 is a permutation group and the elements of formula_1 are represented as permutations of formula_4. A permutation representation is equivalent to an action of formula_1 on the set formula_4:
See the article on group action for further details.
Linear permutation representation.
If formula_1 is a permutation group of degree formula_15, then the permutation representation of formula_1 is the linear representation of formula_1
which maps formula_19 to the corresponding permutation matrix (here formula_20 is an arbitrary field). That is, formula_1 acts on formula_22 by permuting the standard basis vectors.
This notion of a permutation representation can, of course, be composed with the previous one to represent an arbitrary abstract group formula_1 as a group of permutation matrices. One first represents formula_1 as a permutation group and then maps each permutation to the corresponding matrix. Representing formula_1 as a permutation group acting on itself by translation, one obtains the regular representation. 
Character of the permutation representation.
Given a group formula_1 and a finite set formula_4 with formula_1 acting on the set formula_4 then the character formula_30 of the permutation representation is exactly the number of fixed points of formula_4 under the action of formula_32 on formula_4. That is formula_34 the number of points of formula_4 fixed by formula_32. 
This follows since, if we represent the map formula_32 with a matrix with basis defined by the elements of formula_4 we get a permutation matrix of formula_4. Now the character of this representation is defined as the trace of this permutation matrix. An element on the diagonal of a permutation matrix is 1 if the point in formula_4 is fixed, and 0 otherwise. So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of formula_4. 
For example, if formula_42 and formula_43 the character of the permutation representation can be computed with the formula formula_34 the number of points of formula_4 fixed by formula_46.
So 