Cohomology Elements Of Character Triples

Let G be a finite group. Suppose N (?) G and θ ∈ Ittg(N). By some technique, for example, projective repressentation, the theory of coho-mology and so on, we difine a cohomology element ω(θ) in H~2(G/N, C) associated with a character triple (G,N,θ). We use it to describe the extendibility obstruction to 0, i.e., 9 is extendible to G iff ω(θ) = 1.In this paper, we first prove that the map ω preserves products of characters whenever they are well defined, i.e., if 99' G Ittg(N), then ω(θθ') — ω(θ)ω(θ') . We use this result to study the property of p-rational character, the extendibility of fully factored characters of normal solvable subgroups. Secondly, we study the isomorphism problem of two character triples and give some sufEcient conditions for the isomorphism of two character triples, in particular, we obtain a result that (G, N, θ) is isomorphic to (G, N, ψ) whenever ω(θ) = ω(ψ).The following are main results:Theorem 1.2.1 Let AT be a normal subgroup of G and ω : Ivtg(N) - H2(G/N,C*) be the obstruction map.(1) Let θ,θ' ∈ IrrG(N) be such that 99' irreducible. Then ω(θθ') = ω(θ)ω(θ'). In particular, the restriction of obstruction mapω:Ittg(N) -> H2(G/N,C*)is exactly the homomorphism of group.(2) Let θ∈ IrrG(N). Then ω(det θ) = ω(θ)θ(1).(3) Let θ∈ IrrG(N). Then o(ω(θ)) divides (θ(1)o(θ), |G/N|).Theorem 1.2.6 Let N be a normal subgroup of G. Suppose...