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 .