A representation is faithful when its kernel is trivial.
If are the complex irreducible representations of a finite group and , decomposition of the regular representation gives
By Maschke's theorem, the complex regular representation of a finite group is a direct sum of irreducibles. Its kernel is the intersection of their kernels. If the group is simple, each kernel is either trivial or the whole group; faithfulness of the regular representation therefore forces at least one irreducible constituent to be faithful.
For a -cycle , each with is conjugate to . In a faithful complex representation, has a nontrivial th-root eigenvalue , and similarity of and puts all values in its spectrum. Hence the representation has dimension at least .
Commuting complex involutions are simultaneously diagonalizable, so can act faithfully in dimension only if . A group containing for arbitrarily large , such as the full permutation group of , has no faithful finite-dimensional complex representation.
Let be the real upper unitriangular three-by-three group, let be its centre, and let be the integer subgroup of . Every finite-dimensional continuous complex representation of kills the central circle , and hence is not faithful.
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