The spectral sequence of a split extension and the cohomology of an extraspecial group of order p3 and exponent p

Abstract

Let (Er,dr) be the Lyndon-Hochschild-Serre (LHS) spectral sequence associated to a split extension 1 → H → G → G/H → 1 of finite groups with coefficients in a field k. We prove a version of a theorem of Charlap and Vasquez which gives an explicit formula for d2. We then apply this to the case where p is an odd prime, k has characteristic p, G is extraspecial of order p3 and exponent p, and H is elementary abelian of order p2. We calculate the terms of the spectral sequence in this case and prove E3 = E∞ (and if p = 3, E2 = E∞).