On Clifford theory with Galois action

الأصل · EN

Let G be a finite group, N a normal subgroup of G and θ∈ IrrN. Let F be a subfield of the complex numbers and assume that the Galois orbit of θ over F is invariant in G. We show that there is another triple (G₁,N₁,θ₁) of the same form, such that the character theories of G over θ and of G₁ over θ₁ are essentially "the same" over the field F and such that the following holds: G₁ has a cyclic normal subgroup C contained in N₁, such that θ₁=λⁿ¹ for some linear character λ of C, and such that N₁/C is isomorphic to the (abelian) Galois group of the field extension F(λ)/F(θ₁). More precisely, "the same" means that both triples yield the same element of the Brauer-Clifford group BrCliff(G,F(θ)) defined by A. Turull.