Computing Whitehead groups using genetic bases

TL;DR

This paper computes Whitehead groups of certain metacyclic p-groups by combining group theory results and genetic bases, proposing a conjecture and providing explicit torsion group structures.

Contribution

It introduces methods to compute Whitehead groups of metacyclic p-groups using genetic bases and presents a conjecture for the torsion part of specific groups.

Findings

Torsion part of Whitehead group of C_{p^n}×C_{p^n} conjectured

Torsion part of Whitehead group of semidirect product C_{p^{n-1}} ⋉ C_p is C_p^{(n-2)(p-1)}

Techniques applicable to all abelian p-groups and many other p-groups for odd p

Abstract

We combine results about Whitehead groups of finite groups with results about genetic bases of finite $p$-groups to compute the Whitehead groups of some metacyclic $p$-groups. Let $C_{p^n}$ denote a cyclic group of order $p^n$ for $p$ an odd prime and $n$ a positive integer. We present a conjecture for the torsion part of the Whitehead group of $C_{p^n}\times C_{p^n}$ and we show that torsion part of the Whitehead group of the semidirect product of $C_{p^{n-1}}$ by $C_p$ is isomorphic to $C_p^{(n-2)(p-1)}$. The techniques we use can be applied to any abelian $p$-group and to many other $p$-groups for $p$ odd.