Criterions for $U_1(\mathbb{Z}G)=G$ i.e. units to be trivial in $\mathbb{Z}G$

Question

Notation- $U_1(\mathbb{Z}G)$ here means units of $\mathbb{Z}G$ with augmentation $1$, i.e. $u=\sum_{g\in G} a(g)g \in U(\mathbb{Z}G)$ such that $\sum_{g \in G} a(g)=1$

1) I have done theorem by Higman which states that If $G$ is a torsion group then $U_1(\mathbb{Z}G)=G$ iff $G$ is either an abelian group of exponent $1,2,3,4$ or $6$ or a Hamiltonian $2$- group.

2) Another criterion I have done is that if $G \le U_1(\mathbb{Z}G)$ and [$U_1(\mathbb{Z}G):G $] is finite then $U_1(\mathbb{Z}G)=G$.

What are other interesting/ nontrivial criterion for units in $\mathbb{Z}G$ to be trivial? References are welcome too.

Answer 1

In the book An Introduction to Group Rings by Milies and Sehgal there is in chapter 8.2 the following exercise:

(Parmenter and Polcino Milies [116]) Let $G$ be a finite group. Prove that the following conditions are equivalent.

$\phantom{\text{(iiii)}}\llap{\text{(i)}}\;$ $\mathcal{U}(\mathbf{Z}G)$ is nilpotent.
$\phantom{\text{(iiii)}}\llap{\text{(ii)}}\;$ $\mathcal{U}(\mathbf{Z}G)$ is an FC group.
$\phantom{\text{(iiii)}}\llap{\text{(iii)}}\;$ $T\mathcal{U}(\mathbf{Z}G)$ (the set of elements of finite order in $\mathcal{U}(\mathbf{Z}G)$) is a subgroup.
$\phantom{\text{(iiii)}}\llap{\text{(iv)}}\;$ $G$ is either abelian or a Hamiltonian $2$-group.
$\phantom{\text{(iiii)}}\llap{\text{(v)}}\;$ $\mathcal{U}(\mathbf{Z}G)=\pm G$.

[116] M.M. Parmenter and C. Polcino Milies, Group rings whose units form a nilpotent or FC group, Proc. Amer. Math. Soc. 68, 2 (1978), 247-248

You might also like some of the other exercises in this chapter.