In mathematics, the term permutation representation of a (typically finite) group
can refer to either of two closely related notions: a representation of
as a group of permutations, or as a group of permutation matrices.
The term also refers to the combination of the two.
A permutation representation of a group
is a permutation group and the elements of
[1] A permutation representation is equivalent to an action of
: See the article on group action for further details.
is a permutation group of degree
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
as a group of permutation matrices.
as a permutation group and then maps each permutation to the corresponding matrix.
as a permutation group acting on itself by translation, one obtains the regular representation.
and a finite set
acting on the set
of the permutation representation is exactly the number of fixed points of
This follows since, if we represent the map
with a matrix with basis defined by the elements of
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
So we can conclude that the trace of the permutation matrix is exactly equal to the number of fixed points of
the character of the permutation representation can be computed with the formula
So This abstract algebra-related article is a stub.