The spectral sequence of a split extension and the cohomology of an extraspecial group of order \(p^ 3\) and exponent \(p\)
For an extension of finite groups \(1\to H\to G\to G/H\to 1\), a version of a result of Charlap-Vasquez is proved giving an explicit formula for the differential \(d_2\) in the spectral sequence of the extension with coefficients in a field \(k\). This is then applied to the situation where \(k\) is a field of odd characteristic \(p\), \(G\) an extraspecial group of order \(p^3\) and exponent \(p\), and where \(H\) is elementary abelian of order \(p^2\). In this case, the spectral sequence is calculated and it is shown that, for \(p=3\), \(E_2=E_\infty\) and, for \(p\geq 3\), \(E_3=E_\infty\). Reviewer's remark: The author raises the question whether there is a way to calculate the differential \(d_2\) different from the Charlap-Vasquez technique. Such a technique is in fact provided by homological perturbation theory, see e.g. the reviewer's papers [J. Algebra 126, No. 2, 348-399, 400-450 (1989; Zbl 0696.55024, Zbl 0696.55025)] where in Section 8 the cohomology of the group \(G\) (and that of related ones) has been studied; so far, the differential under discussion has not been calculated by homological perturbation theory.
scientific article; zbMATH DE number 858642
scientific article