order, the values of are roots of unity. For example, let be a nontrivial linear representation. Since is a group homomorphism, it has to satisfy Because generates is determined by its value on And as is nontrivial, Thus, we achieve the result that the image of under has to be a nontrivial subgroup of the group which consists of the fourth roots of unity. In other words, has to be one of the following three maps: Let and let be the group homomorphism defined by: In this case is a linear representation of of degree Permutation representation Let be a finite set and let be a group acting on Denote by the group of all permutations on with the composition as group multiplication. A group acting on a finite set is sometimes considered sufficient for the definition of the permutation representation. However, since we want to construct examples for linear representations - where groups act on vector spaces instead of on arbitrary finite sets - we have to proceed in a different way. In order to construct the permutation representation, we need a vector space with A basis of can be indexed by the elements of The permutation representation is the group homomorphism given by for all All linear maps are uniquely defined by this property. Example. Let and Then acts on via The associated linear representation is with for Left- and right-regular representation Let be a group and be a vector space of dimension with a basis indexed by the elements of The left-regular representation is a special case of the permutation representation by choosing This means for all Thus, the family of images of are a basis of The degree of the left-regular representation is equal to the order of the group. The right-regular representation is defined on the same vector space with a similar homomorphism: In the same way as before is a basis of Just as in the case of the left-regular representation, the degree of the right-regular representation is equal to the order of Both representations are isomorphic via For this reason they are not always set apart, and often referred to as "the" regular representation. A closer look provides the following result: A given linear representation is isomorphic to the left-regular representation if and only if there exists a such that is a basis of Example. Let and with the basis Then the left-regular