faithful representation

A faithful representation is one that loses no information about the group: distinct group elements always act by distinct linear maps. Faithfulness is what lets you 'see' the abstract group concretely as a group of matrices, with no collapsing — it is an honest embedding of the group into a matrix group.

Formally, a representation rho: G -> GL(V) is faithful if rho is injective as a homomorphism, equivalently if its kernel is trivial: rho(g) = identity forces g = e. Since the kernel of any representation is a normal subgroup of G, a representation is faithful exactly when that normal subgroup is {e}. The trivial representation, by contrast, is maximally unfaithful — its kernel is all of G — except for the trivial group.

Every finite group has a faithful representation: the regular representation is faithful, because if g acts as the identity on the group algebra then g times e = e forces g = e. More economically, a faithful complex representation need not be irreducible, and the minimal degree of a faithful representation is a subtle invariant; for instance abelian groups that are not cyclic admit no faithful irreducible representation, since irreducible complex representations of abelian groups are one-dimensional and hence have cyclic image.