of to the subgroup If there is no danger of confusion, we might use only or in short The notation or in short is also used to denote the restriction of the representation of onto Let be a function on We write or shortly for the restriction to the subgroup It can be proven that the number of irreducible representations of a group (or correspondingly the number of simple –modules) equals the number of conjugacy classes of A representation is called semisimple or completely reducible if it can be written as a direct sum of irreducible representations. This is analogous to the corresponding definition for a semisimple algebra. For the definition of the direct sum of representations please refer to the section on direct sums of representations. A representation is called isotypic if it is a direct sum of pairwise isomorphic irreducible representations. Let be a given representation of a group Let be an irreducible representation of The –isotype of is defined as the sum of all irreducible subrepresentations of isomorphic to Every vector space over can be provided with an inner product. A representation of a group in a vector space endowed with an inner product is called unitary if is unitary for every This means that in particular every is diagonalizable. For more details see the article on unitary representations. A representation is unitary with respect to a given inner product if and only if the inner product is invariant with regard to the induced operation of i.e. if and only if holds for all A given inner product can be replaced by an invariant inner product by exchanging with Thus, without loss of generality we can assume that every further considered representation is unitary. Example. Let be the dihedral group of order generated by which fulfil the properties and Let be a linear representation of defined on the generators by: This representation is faithful. The subspace is a –invariant subspace. Thus, there exists a nontrivial subrepresentation with Therefore, the representation is not irreducible. The mentioned subrepresentation is of degree one and irreducible. The complementary subspace of is –invariant as well. Therefore, we obtain the subrepresentation with This subrepresentation is also irreducible. That means, the original representation is completely reducible: Both subrepresentations are isotypic and are the two only non-zero isotypes of The representation is unitary with regard to the standard inner product on because and are unitary.