 Cohomological dimension at a prime

The value ${\mathrm{cd}}_p(G)$ is the least $n$ such that all cohomology in degrees greater than $n$ vanishes for every discrete $p$-primary torsion module. Vanishing in degree $n+1$ for all such modules suffices by dimension shifting. Dimension does not increase on closed subgroups. For a field, use its absolute Galois group.