Group cohomology of cyclic group:Z2

This article gives specific information, namely, group cohomology, about a particular group, namely: cyclic group:Z2.
View group cohomology of particular groups | View other specific information about cyclic group:Z2

Classifying space and corresponding chain complex

The classifying space of the cyclic group of order two is $\mathbb{R}{\mathbb{P}}^{\mathrm{\infty }}$, viz., countable-dimensional real projective space (read more about this space as a topological space on the Topology Wiki).

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

$\dots \stackrel{\cdot 0}{\to }\mathbb{Z}\stackrel{\cdot 2}{\to }\mathbb{Z}\stackrel{\cdot 0}{\to }\mathbb{Z}\to \dots \stackrel{\cdot 2}{\to }\mathbb{Z}\stackrel{\cdot 0}{\to }\mathbb{Z}$

where the subscript for the last written entry is $0$, and hence the multiplication by 2 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 countable-dimensional real projective space at the Topology Wiki

Over the integers

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

$H_p(\mathbb{Z}/2\mathbb{Z};\mathbb{Z})=\{\begin{array}{cc}\mathbb{Z}/2\mathbb{Z},& p=1,3,5,\dots \\ 0,& p=2,4,6,\dots \\ \mathbb{Z},& p=0\\ \end{array}$

Over an abelian group or module $M$

The homology groups with coefficients in a module $M$ over a ring $R$ are given by:

$H_p(\mathbb{Z}/2\mathbb{Z};M)=\{\begin{array}{cc}M/2M,& p=1,3,5,\dots \\ T,& p=2,4,6,\dots \\ M,& p=0\\ \end{array}$

where $T$ is the 2-torsion submodule of $M$, i.e., the submodule of $M$ comprising elements whose double is 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 2-divisible, i.e., every element of $M$ has a unique half. This includes the case that $M$ is a field of characteristic not 2.	all zero groups	all zero groups
$M$ is 2-torsion-free, i.e., no nonzero element of $M$ doubles to zero	unclear	all zero groups
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M=\mathbb{Q}/\mathbb{Z}$	all zero groups	unclear
$M=\mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number	all isomorphic to $\mathbb{Z}/2\mathbb{Z}$	all isomorphic to $\mathbb{Z}/2\mathbb{Z}$
$M$ is a finite abelian group	all isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^r$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$	all isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^r$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$

Note that the third case, where $M$ is 2-divisible but not necessarily uniquely so, cannot arise if $M=R$ and it is a unital ring. So when taking coefficients over a unital ring, there is no need to distinguish between 2-divisibility and unique 2-divisibility.

Cohomology groups and cohomology ring

To look at the same material from a topological/algebraic topology perspective, check out the cohomology of countable-dimensional real projective space at the Topology Wiki

Groups over the integers

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

$H^p(\mathbb{Z}/2\mathbb{Z};\mathbb{Z})=\{\begin{array}{cc}0,& p=1,3,5,\dots \\ \mathbb{Z}/2\mathbb{Z},& p=2,4,6,\dots \\ \mathbb{Z},& p=0\\ \end{array}$

Basically, the even/odd role gets interchanged. This is because in the cochain complex, the arrows are all pointing in the reverse direction.

Cohomology ring

The cohomology groups over the integers come with a ring structure. The structure of the ring is $\mathbb{Z}[x]/⟨2x⟩$. It is almost the same as $(\mathbb{Z}/2\mathbb{Z})[x]$ but the key difference is that the constant terms can vary over all of $\mathbb{Z}$. The identification is as follows: $x^k$ is the unique nonzero element in $H^{2k}(\mathbb{Z}/2\mathbb{Z};\mathbb{Z})$.

Over a module $M$ over a ring $R$

The cohomology groups with coefficients in a module $M$ over a ring $R$ are given by:

$H^p(\mathbb{Z}/2\mathbb{Z};M)=\{\begin{array}{cc}T,& p=1,3,5,\dots \\ M/2M,& p=2,4,6,\dots \\ M,& p=0\\ \end{array}$

where $T$ is the 2-torsion submodule of $M$, i.e., the submodule of $M$ comprising elements whose double is zero.

In particular, we see the following cases:

Case on $R$ or $M$	Conclusion about odd-indexed cohomology groups, i.e., $H^p,p=1,3,5,\dots$	Conclusion about even-indexed cohomology groups, i.e., $H^p,p=2,4,6,\dots$	Conclusion about cohomology ring when $M=R$
$M$ is uniquely 2-divisible, i.e., every element of $M$ has a unique half	all zero groups	all zero groups	The ring is just $R$, i.e., only constant polynomials
$M$ is 2-torsion-free, i.e., no nonzero element of $M$ doubles to zero	all zero groups	unclear	The ring is $R[x]/⟨2x⟩$. Note that $R=\mathbb{Z}$ is a special example of this
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M=\mathbb{Q}/\mathbb{Z}$	unclear	all zero groups	Not applicable, see note below table
$M=\mathbb{Z}/2^n\mathbb{Z}$	all isomorphic to $\mathbb{Z}/2\mathbb{Z}$	all isomorphic to $\mathbb{Z}/2\mathbb{Z}$	What do we get?
$M$ is a finite abelian group	all isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^r$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$	all isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^r$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$	What do we get?

Second cohomology group and extensions

Schur multiplier

The Schur multiplier, defined as the second cohomology group for trivial group action $H^2(G,{\mathbb{C}}^{\ast })$ and also as the second homology group $H_2(G,\mathbb{Z})$, is the trivial group.

In other words, cyclic group:Z2 is a Schur-trivial group. See also cyclic implies Schur-trivial.

Second cohomology groups for trivial group action

As noted above, if $M$ is a finite abelian group, all the cohomology groups (for trivial group action) $H^p(\mathbb{Z}/2\mathbb{Z};M)$ are isomorphic to $(\mathbb{Z}/2\mathbb{Z}{)}^r$ where $r$ is the rank (i.e., minimum size of generating set) for the 2-Sylow subgroup of $M$. In particular, this is also true for the second cohomology group for trivial group action.

The corresponding extensions to the elements of this second cohomology group are all abelian group extensions. We list some cases below:

Case for $M$	Isomorphism class of second cohomology group	Link to page	Short description of extensions
$M$ has odd order	trivial group	--	The only extension is a direct product of $M$ and the cyclic group of order two.
$M=\mathbb{Z}/2\mathbb{Z}$, i.e., cyclic group:Z2	cyclic group:Z2	second cohomology group for trivial group action of Z2 on Z2	The extensions are Klein four-group (for zero element of cohomology group) and cyclic group:Z4 (for nonzero element)
$M=\mathbb{Z}/4\mathbb{Z}$, i.e., cyclic group:Z4	cyclic group:Z2	second cohomology group for trivial group action of Z2 on Z4	The extensions are direct product of Z4 and Z2 (for zero element of cohomology group) and cyclic group:Z8 (for nonzero element)
$M=\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, i.e., Klein four-group	Klein four-group	second cohomology group for trivial group action of Z2 on V4	The extensions are elementary abelian group:E8 (for zero element of cohomology group) and direct product of Z4 and Z2 (three copies; different versions for each of the nonzero elements)