Group cohomology of finite cyclic groups

Classifying space and corresponding chain complex
The classifying space of a finite cyclic group of order $$n$$ is a lens space (Read more about the lens space as a topological space on the Topology Wiki).

A chain complex that can be used to compute the homology and cohomology for the classifying space and hence also for the group is:

$$\dots \stackrel{\cdot 0}{\to} \mathbb{Z} \stackrel{\cdot n}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z} \to \dots \stackrel{\cdot n}{\to} \mathbb{Z} \stackrel{\cdot 0}{\to} \mathbb{Z}$$

where the subscript for the last written entry is $$0$$, and hence the multiplication by $$n$$ maps arise from even to odd subscripts and the multiplication by zero maps arise from odd to even subscripts.

Over the integers
The homology groups with coefficients in the ring of integers $$\mathbb{Z}$$ are given as follows:

$$H_p(\mathbb{Z}/n\mathbb{Z};\mathbb{Z}) = \left\lbrace\begin{array}{rl}\mathbb{Z}/n\mathbb{Z}, &p = 1,3,5,\dots\\0, & p = 2,4,6, \dots \\ \mathbb{Z},& p = 0\\\end{array}\right.$$

Over an abelian group
The homology groups with coefficients in an abelian group $$M$$ (which we may treat as a module over a unital ring $$R$$, which could be $$\mathbb{Z}$$ or something else) are given by:

$$H_p(\mathbb{Z}/n\mathbb{Z};M) = \left\lbrace\begin{array}{rl} M/nM, & p=1,3,5,\dots\\ T, & p = 2,4,6, \dots \\ M, & p = 0\\\end{array}\right.$$

where $$T$$ is the $$n$$-torsion submodule of $$M$$, i.e., the submodule of $$M$$ comprising elements which, when multiplied by $$n$$, give zero.

In particular, we see the following cases:

Cohomology groups
The cohomology groups with coefficients in the ring of integers $$\mathbb{Z}$$ are given as follows:

$$H^p(\mathbb{Z}/n\mathbb{Z};\mathbb{Z}) = \left\lbrace\begin{array}{rl}\mathbb{Z}/n\mathbb{Z}, &p = 2,4,6,\dots\\0, & p = 1,3,5, \dots \\ \mathbb{Z},& p = 0\\\end{array}\right.$$

Over an abelian group
The cohomology groups with coefficients in an abelian group $$M$$ (which we may treat as a module over a unital ring $$R$$, which could be $$\mathbb{Z}$$ or something else) are given by:

$$H^p(\mathbb{Z}/n\mathbb{Z};M) = \left\lbrace\begin{array}{rl} M/nM, & p=2,4,6,\dots\\ T, & p = 1,3,5, \dots \\ M, & p = 0\\\end{array}\right.$$

where $$T$$ is the $$n$$-torsion submodule of $$M$$, i.e., the submodule of $$M$$ comprising elements which, when multiplied by $$n$$, give zero.