The cohomological and homological dimension of a group do not agree in general. For example, the homological dimension of the group $Z[\frac{1}{2}]$ is one, while its cohomological dimension is 2. However, if |G| is countable, then cohomological dimension is either equal to the homological dimension or one dimension greater.
Also , cohomological dimension of a group is defined by the highest dimension n such that $H^n(G,M)$ is nonzero, where M is a G-module. It is important to use a G-module instead of the usual cofficient. For example, the cohomological dimension for any nontrivial knot group is 2, while its cohomology with $Z$ cofficient, or any G-module with trivial group action is always the same as the cohomology of $S^1$.
The cohomological and homological dimension of a group do not agree in general. For example, the homological dimension of the group $Z[\frac{1}{2}]$ is one, while its cohomological dimension is 2. However, |G| is countable, then cohomological dimension is either equal to the homological dimension or one dimension greater.
Also , cohomological dimension of a group is defined by the highest dimension n such that $H^n(G,M)$ is nonzero, where M is a G-module. It is important to use a G-module instead of the usual cofficient. For example, the cohomological dimension for any nontrivial knot group is 2, while its cohomology with $Z$ cofficient, or any G-module with trivial group action is always the same as the cohomology of $S^1$.