The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the category of $G$-modules to the category of Abelian groups sending each module $M$ to its subgroup consisting of all elements of $M$ which are fixed by $G$ action.

Now we fix a natural number $n\in \mathbb{N}$ and do an obvious generalization of the above construction:

Instead of the above left exact functor $FIX$ we consider the functor $P_n$ which send a $G$-midule $M$ to the following subgroup of $M$: $$P_n(M)=\{x\in M \mid g.x= nx\}$$.

In this way what would be the corresponding derived functor?What kind of cohomology theory would appear? Is this a trivial generalization of classical "Group Cohomology"?If not, what would be a topological analogy?

The later question is motivated by the fact that the group cohomology corresponds to singular cohomology of the corresponding Eilenberg-Mclane space)

The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the category of $G$-modules to the category of Abelian groups sending each module $M$ to its subgroup consisting of all elements of $M$ which are fixed by $G$ action.

Now we fix a natural number $n\in \mathbb{N}$ and do an obvious generalization of the above construction:

Instead of the above left exact functor $FIX$ we consider the functor $P_n$ which send a $G$-midule $M$ to the following subgroup of $M$: $$P_n(M)=\{x\in M \mid g.x= nx, \forall g \in G\}$$.

In this way what would be the corresponding derived functor?What kind of cohomology theory would appear? Is this a trivial generalization of classical "Group Cohomology"?If not, what would be a topological analogy?

The later question is motivated by the fact that the group cohomology corresponds to singular cohomology of the corresponding Eilenberg-Mclane space)