Representations of Finite Group Extensions via Projective Representations

Abstract

AbstractIn this chapter we give a formulation of the little group method (cf. Theorem 2.26) for arbitrary group extensions, that is, we do not assume existence of an extension of \(\sigma \in \widehat {N}\) to its inertia group I G(σ). We also give a sufficient condition for the existence of such an extension. All these facts require a suitable cohomological machinery. It is due to Mackey (who actually proved more general results for topological groups), but some aspects of it were anticipated by Schur and others within the theory of projective representations; see Berkovich and Zhmud (Characters of finite groups. Part 1, Translations of Mathematical Monographs, vol 172. American Mathematical Society, Providence, 1998), Huppert (Character Theory of Finite Groups, De Gruyter Expositions in Mathematics, vol 25. Walter de Gruyter, 1998), and Isaacs (Character theory of finite groups, Corrected reprint of the 1976 original [Academic Press, New York]. Dover Publications, New York, 1994). Actually, we follow Mackey’s approach as described in Fell and Doran (Pure and Applied Mathematics, vol. 126. Academic Press, Boston, 1988); see also Tucker (Am J Math 84:400–420).