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

TL;DR

This paper computes the Whitehead group of almost extra-special p-groups for odd primes, introduces a deflation map, and provides new proofs for known structures of SK_1(ZP) in certain cases.

Contribution

It offers a complete description of the Whitehead group for almost extra-special p-groups and introduces a surjective deflation map for Cl_1(ZP).

Findings

Computed Cl_1(ZP) for all finite p-groups P.

Introduced a surjective deflation map Cl_1(ZP) → Cl_1(Z(P/N)).

Provided a new proof for the structure of SK_1(ZP) when P is elementary abelian.

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) $\rightarrow$ 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 p-group.