Some criteria concerning the rational vanishing of Whitehead groups

TL;DR

This paper explores conditions under which Whitehead groups of finite groups vanish or not, providing examples and implications for larger groups containing these finite groups as normal subgroups.

Contribution

It offers new examples and criteria related to the rational vanishing of Whitehead groups for finite groups, extending previous theoretical results.

Findings

Examples of finite groups with nonzero Whitehead group tensor rank

Criteria for the vanishing or non-vanishing of Whitehead groups

Implications for larger groups containing these finite groups as normal subgroups

Abstract

We give several examples of finite groups $G$ for which the rank of the tensor product $\mathbb{Z} \otimes_{\mathbb{Z}\mathrm{Aut}(G)}$ Wh$(G)$ is or is not zero. This is motivated by an earlier theorem of the first author, which implies as a special case that when this group has nonzero rank, the Whitehead group of any other group (finite or infinite) that contains $G$ as a normal subgroup is rationally nontrivial.