# Faithful representation

In mathematics, especially in the 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*).

In more abstract language, this means that the group homomorphism

- ρ:
*G*→*GL*(*V*)

is injective.

*Caveat:* While representations of *G* over a field *K* are *de facto* the same as -modules (with 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 -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.

## Properties

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*.

## References

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