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
Consider for example the natural representation of the symmetric group Sn in n dimensions by permutation matrices, which is certainly faithful.
while the n × n matrices form a vector space of dimension n2.
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.