A representation is faithful when its kernel is trivial.
The character of the regular representation of a finite group is at the identity and zero elsewhere. Left multiplication permutes its group-indexed basis, and a nonidentity element fixes none of those basis vectors. Thus each complex irreducible representation occurs with representation multiplicity equal to its dimension, by the character inner product.
The permutation matrices of the three displayed double transpositions, together with , form a faithful representation of the Klein four-group. They satisfy and . Every nonidentity element fixes no basis vector, so this is its four-dimensional regular representation.
The constant vector spans a nonzero proper invariant subspace, proving reducibility. More completely, the columns ofform a common orthonormal basis of eigenvectors. Direct multiplication gives and . Thus the representation splits into the four distinct real one-dimensional characters, each once. The basis change is a normalized Hadamard matrix.
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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