Group cohomology of finite cyclic groups

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This article gives specific information, namely, group cohomology, about a family of groups, namely: finite cyclic group.
View group cohomology of group families | View other specific information about finite cyclic group

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.

Homology groups

To look at the same material from a topological/algebraic topology perspective, check out the homology of lens space at the Topology Wiki

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:

Case on R or M Conclusion about odd-indexed homology groups, i.e., H_p, p = 1,3,5,\dots Conclusion about even-indexed homology groups, i.e., H_p, p = 2,4,6,\dots
M is uniquely n-divisible, i.e., every element of M can be uniquely divided by n. This includes the case that M is a field of characteristic not dividing n. all zero groups all zero groups
M is n-torsion-free, i.e., no nonzero element of M doubles to zero unclear all zero groups
M is n-divisible, but not necessarily uniquely so, e.g., M = \mathbb{Q}/\mathbb{Z} all zero groups unclear

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.