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 Cl 1 (ZP ) of SK 1 (ZP ), in terms of a genetic basis of P . We also introduce a deflation map Cl 1 (ZP ) → Cl 1 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 SK 1 (ZP ), when P is an elementary abelian pgroup.