Cohomological dimension of a group

Remarks

Equivalently, $\mathrm{cd}(G)$ is the smallest $n$ such that $\Z$, with trivial action, has a projective resolution of length $n$ over the group ring $\Z[G]$. For a subgroup $H$ of $G$, $\mathrm{cd}(H) \le \mathrm{cd}(G)$. The only group with $\mathrm{cd}(G) = 0$ is the trivial group.

Example: $\mathrm{cd}(\Z) = 1$. The resolution $0 \to \Z[t, t^{-1}] \xrightarrow{\ t - 1\ } \Z[t, t^{-1}] \to \Z \to 0$ has length $1$, and $H^1(\Z, \Z) \cong \Z$ is not zero. More generally, $\mathrm{cd}(\Z^n) = n$, and a nontrivial free group has cohomological dimension $1$.

Non-example: a nontrivial finite cyclic group $C_m$ has $H^{2k}(C_m, \Z) \cong \Z/m\Z \ne 0$ for every $k \ge 1$, so no integer $n$ works and $\mathrm{cd}(C_m) = \infty$. Since a group containing $C_m$ has dimension at least that of $C_m$, every group of finite cohomological dimension is torsion-free.