Kernels of projective representations of finite groups (Q1190040)

We will augment and refine some of the results of an earlier article [Zap. Mat. Otd. Fiz. Mat. Fak. KhGU Khar'k Mat. Ob., IV. Ser. 26, 333-372 (1960)]. Recall the definition of an \(H\)-multiplier of a finite group \(G\), a notion that was introduced there: Let \(\Pi\) be a class of associated factor systems of the complex projective representations of the group \(G\). (In other words, \(\Pi\) is an element of the Schur multiplier \(M(G)\) of the group \(G\).) A normal subgroup \(H\) of a group \(G\) is called a \(\Pi\)-kernel of \(G\) if it is the kernel of a complex projective representation of \(G\) belonging to the factor system \(\pi \in \Pi\). The set \(M_ H(G) = \{\Pi \in M(G)\mid H \text{ is a }\Pi\text{- kernel}\}\) (which, as we can show, is a subgroup of \(M(G)\)) is called the \(H\)-multiplier of the group \(G\). One of the fundamental results of the present article is Theorem 2.12, which states that \(M_ H(G) \cong M(G/H)/N\), where \(N \cong H\cap G'/[H,G]\) (where \(G'\) is commutator of \(G\), and \([H,G]\) is the commutator of \(H\) and \(G\)). The proof of Theorem 2.12 is based on examination of a particular sequence of groups and homomorphisms. Let \(\text{Lin}(G)\) be the group of complex linear characters of \(G\), and let \(\text{Lin}_{\text{inv}}(H)\) be the group of \(G\)-invariant characters of the subgroup \(H\). It is natural to define the homomorphisms \(\phi : \text{Lin}(G) \to \text{Lin}_{\text{inv}}(H)\), \(\tau: \text{Lin}_{\text{inv}}(H) \to M(G/H)\), \(\sigma: M(G/H) \to M_ H(G)\). Theorem 2.12 is a corollary of Theorem 2.8, from which it follows that the sequence \(\text{Lin}(G) @>\varphi>> \text{Lin}_{\text{inv}}(H) @>\tau>> M(G/H) @>\sigma>> M_ H(G) \to 1\) is exact.