is surjective, provided that X is the family of all elementary subgroups. Here a group H is elementary if there is some prime p such that H is the direct product of a cyclic group of order prime to and a –group. In other words, every character of is a linear combination with integer coefficients of characters induced by characters of elementary subgroups. The elementary subgroups H arising in Brauer's theorem have a richer representation theory than cyclic groups, they at least have the property that any irreducible representation for such H is induced by a one-dimensional representation of a (necessarily also elementary) subgroup . (This latter property can be shown to hold for any supersolvable group, which includes nilpotent groups and, in particular, elementary groups.) This ability to induce representations from degree 1 representations has some further consequences in the representation theory of finite groups. Real representations For proofs and more information about representations over general subfields of please refer to [2]. If a group acts on a real vector space the corresponding representation on the complex vector space is called real ( is called the complexification of ). The corresponding representation mentioned above is given by for all Let be a real representation. The linear map is -valued for all Thus, we can conclude that the character of a real representation is always real-valued. But not every representation with a real-valued character is real. To make this clear, let be a finite, non-abelian subgroup of the group Then acts on Since the trace of any matrix in is real, the character of the representation is real-valued. Suppose is a real representation, then would consist only of real-valued matrices. Thus, However the circle group is abelian but was chosen to be a non-abelian group. Now we only need to prove the existence of a non-abelian, finite subgroup of To find such a group, observe that can be identified with the units of the quaternions. Now let The following two-dimensional representation of is not real-valued, but has a real-valued character: Then the image of is not real-valued, but nevertheless it is a subset of Thus, the character of the representation is real. Lemma. An irreducible representation of is real if and only if there exists a nondegenerate symmetric bilinear form on preserved by An irreducible representation of on a real vector space can become reducible when extending the field to