# Group cohomology of finite cyclic groups

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 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:

where the subscript for the last written entry is , and hence the multiplication by 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 are given as follows:

### Over an abelian group

The homology groups with coefficients in an abelian group (which we may treat as a module over a unital ring , which could be or something else) are given by:

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

In particular, we see the following cases:

Case on or | Conclusion about odd-indexed homology groups, i.e., | Conclusion about even-indexed homology groups, i.e., |
---|---|---|

is uniquely -divisible, i.e., every element of can be uniquely divided by . This includes the case that is a field of characteristic not dividing . | all zero groups | all zero groups |

is -torsion-free, i.e., no nonzero element of doubles to zero | unclear | all zero groups |

is -divisible, but not necessarily uniquely so, e.g., | all zero groups | unclear |

## Cohomology groups

The cohomology groups with coefficients in the ring of integers are given as follows:

### Over an abelian group

The cohomology groups with coefficients in an abelian group (which we may treat as a module over a unital ring , which could be or something else) are given by:

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