The spectral sequence of a split extension and the cohomology of an
extraspecial group of order $p^3$ and exponent $p$

Author: Stephen F. Siegel

Title: The spectral sequence of a split extension and the cohomology
of an extraspecial group of order $p^3$ and exponent $p$

Abstract: Let $(E_r,d_r)$ be the LHS spectral sequence associated to a
split extension $1\ra H\ra G\ra G/H\ra 1$ 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 $d_2$. We then
apply this to the case where $p$ is an odd prime, $k$ has
characteristic $p$, $G$ is extraspecial of order $p^3$ and exponent
$p$, and $H$ is elementary abelian of order $p^2$. We calculate the
terms of the spectral sequence in this case and prove $E_3=E_\infty$
(and if $p=3$, $E_2=E_\infty$).

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This paper will appear in the Journal of Pure and Applied Algebra.
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Author address:
        Stephen F. Siegel
        Department of Mathematics
        Northwestern University
        Evanston, IL 60208-2730
        708-491-5594