The Embedding Problem in Galois Theory

Relative Embedding Problems

“…By [ILF,Theorem 3.15.1], the multiplication of classes in H 2 (η N (Γ), µ p ) corresponds on one hand to function multiplication, and on the other to the Baer product of the group extensions corresponding to the classes, where the amalgamation occurs over Γ. Hence the quotient (inf N/SB c f )(inf N/TB c −1 f ) describes the Baer productΓ of Γ 1 and Γ 2 over Γ.…”

“…We note that several results of [ILF,§3.15] can be viewed as a treatment of abelian relative embedding problems in the case S B = R 1 .…”

“…The Baer product is then a composition law on the classes of equivalent group extensions, under which these classes form an abelian group isomorphic to H 2 (H, B) [ILF,Theorem 3.15.1]. Recall that the inverse class of G, an arbitrary extension of H by B, is given as follows.…”

“…Note that the map g →ḡ −1 is the automorphism of B given by inversion. Definition 1.6 ( [ILF,§3.15]). Let G 1 and G 2 be finite groups which are group extensions of a finite group H by a H-module B, where the injections of B into G 1 and G 2 are fixed.…”

“…First we determine the subextension of T ⊗ R1 T with action η T ⊗ η T fixed by elements (a, a). Following [ILF,lemma in Theorem 3.15.2], the algebra T ⊗ R1 T with action η T ⊗ η T can be written as a sum a∈A T e 1 , where e ηT (g)⊗ηT (g) 1 = e 1 for each g ∈ G; e a = e ηT (a)⊗ηT (1) 1 ; and G acts on T via the first factor in η T ⊗ η T . Since A is abelian, we have, moreover, that…”