What rhymes with faithful representation?
faith·ful rep·re·sen·ta·tion

In mathematics, especially in an area of abstract algebra known as representation theory, a faithful representation ρ of a group G {\displaystyle G} on a vector space V {\displaystyle V} is a linear representation in which different elements g {\displaystyle g} of G {\displaystyle G} are represented by distinct linear mappings ρ ( g ) {\displaystyle \rho (g)} . In more abstract language, this means that the group homomorphism ρ : G → G L ( V ) {\displaystyle \rho :G\to GL(V)} is injective (or one-to-one). Caveat: While representations of G {\displaystyle G} over a field K {\displaystyle K} are de facto the same as K [ G ] {\displaystyle K[G]} -modules (with K [ G ] {\displaystyle K[G]} denoting the group algebra of the group G {\displaystyle G} ), a faithful representation of G {\displaystyle G} is not necessarily a faithful module for the group algebra. In fact each faithful K [ G ] {\displaystyle K[G]} -module is a faithful representation of G {\displaystyle G} , but the converse does not hold. Consider for example the natural representation of the symmetric group S n {\displaystyle S_{n}} in n {\displaystyle n} dimensions by permutation matrices, which is certainly faithful. Here the order of the group is n {\displaystyle n} ! while the n × n {\displaystyle n\times n} matrices form a vector space of dimension n 2 {\displaystyle n^{2}} . As soon as n {\displaystyle n} is at least 4, dimension counting means that some linear dependence must occur between permutation matrices (since 24 > 16 {\displaystyle 24>16} ); this relation means that the module for the group algebra is not faithful.

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