Projective Characters of Finite Groups

We shall refer to a representation with trivial cocyle as a linear representation. The connection between projective representations and linear representations is provided by the following result of Schur (see [2] or [4]). Given a finite group G, there exists a group H, called a representation group of G, such that H has a central subgroup A with (i) A contained in the derived group of H, (ii) H/A ~ G, and (iii) | A | = | H{Q, K*) |. Furthermore, each projeotive representation of G can be 'linearized' by a linear representation of H, in a sense to be made precise later. The purpose of this article is to make use of this connection to define projective characters of G by reference to the characters of linear representations of H; and to investigate the properties of projective characters. Section 1 contains the definition of the projective characters of G and gives some immediate consequences of the definition. There are several choices involved in our definition. In §§ 2 and 3 we investigate the way in which these choices affect the values of the projective characters. Thus in § 3, which is of independent interest, we show the relationship between the characters of the linear irreducible representations of two different