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 $n$ 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.