Chern classes and the extraspecial p-group of order p⁵ and exponent p

Abstract

For p an odd prime, the cohomology ring of the extraspecial p-group of order p ⁵ and exponent p is investigated.A presentation is obtained for the subquotient generated by Chern classes, modulo nilradical.

Moreover, it is proved that, for every extraspecial p-group of exponent p, the top Chern classes of the irreducible representations do not generate the Chern subring modulo nilradical.Finally, a related question about symplectic invariants is discussed, and is solved for Sp₄(F_p).