On the cohomology of central Frattini extensions

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In this paper we provide calculations for the modp cohomology of certain p-groups, using topological methods. More precisely, we look at p-groups G defined as central extensions 1→V→G→W→1 of elementary abelian groups such that G/[G,G]⊗Fp=W and the defining k-invariants span the entire image of the Bockstein. We show that if p>dimV-dimW+1, then the modp cohomology of G can be explicitly computed as an algebra of the form P⊗A where P is a polynomial ring on two-dimensional generators and A is the cohomology of a compact manifold which in turn can be computed as the homology of a Koszul complex. As an application we provide a complete determination of the modp cohomology of the universal central extension 1→H2(W,Fp)→U→W→1 provided p>(n2)+1, where n=dimW. © 2001 Elsevier Science B.V.

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Adem, A., & Pakianathan, J. (2001). On the cohomology of central Frattini extensions. Journal of Pure and Applied Algebra, 159(1), 1–14. https://doi.org/10.1016/S0022-4049(00)00117-1

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