# Faithful representation

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

In more abstract language, this means that the group homomorphism

is injective (or one-to-one).

*Caveat:* While representations of
over a field
are *de facto* the same as
-modules (with
denoting the group algebra of the group
), a faithful representation of
is not necessarily a faithful module for the group algebra. In fact each faithful
-module is a faithful representation of
, but the converse does not hold. Consider for example the natural representation of the symmetric group
in
dimensions by permutation matrices, which is certainly faithful. Here the order of the group is
! while the
matrices form a vector space of dimension
. As soon as
is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since
); this relation means that the module for the group algebra is not faithful.

## Properties

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

(the -th tensor power of the representation ) for a sufficiently high .

## References

Hazewinkel, Michiel, ed. (2001) [1994], "faithful representation", *Encyclopedia of Mathematics*, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4