$\newcommand{\Z}{\mathbf{Z}}$ Let $G$ be a non-abelian group. And let $\Z$ be the ring of integers. Under which condition on the group $G$ can we find a free resolution $F_{\bullet}\rightarrow \Z$ of $\Z$ as left $\Z[G]$-module satisfying the conditions ?

- $F_{\bullet}\rightarrow \Z$ is a finite resolution I.e. each $F_{i}$ is finitely generated free left $\Z[G]$-module. And $F_{i}=0$ for $i$ enough big.
- $F_{\bullet}\rightarrow \Z$ is also a resolution of $\Z[G]$-bimodules. I.e. for each $i$, $F_{i+1}\rightarrow F_{i}$ and $F_{0}\rightarrow \Z$ are maps of $\Z[G]$-bimodule.

In this question each free $\Z[G]$-module is seen as $\Z[G]$-bimodule in the obvious way.

As far as I know, there is a big class of groups satisfying condition 1. I will be happy if one can provide a non-abelian example satisfying 1 and 2. Or a proof that such non-abelian group does not exist.

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