# Faithful representation

In mathematics, especially in an area of abstract algebra known as representation theory, a **faithful representation** ρ of a group G on a vector space V is a linear representation in which different elements g of G are represented by distinct linear mappings *ρ*(*g*).

*Caveat:* While representations of G over a field K are *de facto* the same as *K*[*G*]-modules (with *K*[*G*] denoting the group algebra of the group G), a faithful representation of G is not necessarily a faithful module for the group algebra. In fact each faithful *K*[*G*]-module is a faithful representation of G, but the converse does not hold. Consider for example the natural representation of the symmetric group *S _{n}* in n dimensions by permutation matrices, which is certainly faithful. Here the order of the group is

*n*! while the

*n*×

*n*matrices form a vector space of dimension

*n*

^{2}. As soon as n is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16); this relation means that the module for the group algebra is not faithful.

A representation V of a finite group G over an algebraically closed field K of characteristic zero is faithful (as a representation) if and only if every irreducible representation of G occurs as a subrepresentation of *S ^{n}V* (the n-th symmetric power of the representation V) for a sufficiently high n. Also, V is faithful (as a representation) if and only if every irreducible representation of G occurs as a subrepresentation of

(the n-th tensor power of the representation V) for a sufficiently high n.^{[1]}