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 $ρ(g)$. In more abstract language, this means that the group homomorphism $$\rho: G\to GL(V)$$ is injective (or one-to-one).

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 $n&thinsp;×&thinsp;n$ 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
 * $$V^{\otimes n} = \underbrace{V \otimes V \otimes \cdots \otimes V}_{n\text{ times}}$$

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