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Alonso Perez-Lona (Feb 28 2023 at 17:46):Hi, I am trying to compute the third cohomology group with coefficients in U(1) of a 2-group finite G using the Serre spectral sequence and the fibration B^2A->BG->BC, with all groups finite and A abelian. Part of the computation requires me to know H^3(B^2A,U(1)). According to a paper by EIlenberg and MacLane (On the Groups H(Π, n), II: Methods of Computation; Eq 26.3), this cohomology group is given by the abelian extensions Extabel(A,U(1)), which is the subgroup of H^2(A,U(1)) coming from symmetric cocycles. Simple examples such as A=Z2xZ2 show that H^2(A,U(1)) is not always trivial. My question is, when is the subgroup of abelian extensions Extabel(A,U(1)) trivial, for A a finite abelian group?
John Baez (Feb 28 2023 at 21:44):Have you worked it out for A = Z/2? It seems that would shed a lot of light on the whole situation.
John Baez (Feb 28 2023 at 21:45):So: is there an abelian extension
that does not split?
John Baez (Feb 28 2023 at 21:47):If I had to guess by sheer instinct, I would guess "no".