The Whitehead group of (almost) extra-special p-groups with p odd

Abstract

Let p be an odd prime number. We describe the Whitehead group of all extra-special and almost extra-special p-groups. For this we compute, for any finite p-group P, the subgroup Cl1(ZP) of SK1(ZP), in terms of a genetic basis of P. We also introduce a deflation map Cl1(ZP)→Cl1(Z(P/N)), for a normal subgroup N of P, and show that it is always surjective. Along the way, we give a new proof of the result describing the structure of SK1(ZP), when P is an elementary abelian p-group.