Group cohomology of Klein four-group

This article gives specific information, namely, group cohomology, about a particular group, namely: Klein four-group.
View group cohomology of particular groups | View other specific information about Klein four-group

Classifying space and corresponding chain complex

The classifying space of the Klein four-group is the product space $\mathbb{R}\mathbb{P}^\infty \times \mathbb{R}\mathbb{P}^\infty$, where $\mathbb{R}\mathbb{P}^\infty$ is infinite-dimensional real projective space.

A chain complex that can be used to compute the homology of this space is given as follows:

• The $n^{th}$ chain group is a sum of $n + 1$ copies of $\mathbb{Z}$, indexed by ordered pairs $(i,j)$ where $i + j = n$. In other words, the $n^{th}$ chain group is:

$\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}$

• The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:

• • The map $\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i,j-1)}$ is multiplication by zero if $j$ is odd and is multiplication by two if $j$ is even.
  • The map $\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i-1,j)}$ is multiplication by zero if $i$ is odd and multiplication by two if $i$ is even.

Homology groups

Over the integers

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

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

The first few homology groups are given below:

$p$	$\! 0$	$\! 1$	$\! 2$	$\! 3$	$\! 4$	$\! 5$
$H_p$	$\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
rank of $H_p$ as an elementary abelian 2-group	--	2	1	3	2	4

These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 (see group cohomology of cyclic group:Z2) using the Kunneth formula for group homology. They can also be computed explicitly using the chain complex description above.

Here is the computation using the Kunneth formula for group homology: [SHOW MORE]

We set $G_1 = G_2 = \mathbb{Z}/2\mathbb{Z}$ and $M = \mathbb{Z}$ in the formula.

Case on $p$	Value of $H_i(G_1;M) \otimes H_j(G_2;M)$ where $i + j = p$	Value of $\operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M))$ where $u + v = p -1$	Value of $\bigoplus_{i+j = p} H_i(G_1;M) \otimes H_j(G_2;M)$	Value of $\bigoplus_{u + v = p - 1} \operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M))$	Value of $H_p(G_1 \times G_2;M)$ (sum of preceding two columns
$p = 0$	$\mathbb{Z}$ in case $i = j = 0$	No such cases	$\mathbb{Z}$	0	$\mathbb{Z}$
$p$ odd positive	$\mathbb{Z}/2\mathbb{Z}$ for the case $i = 0, j = p$ and the case $i = p, j = 0$. 0 in all other cases, because for $i + j$ to be odd, at least one of $i$ and $j$ must be even, forcing the corresponding homology group to be 0.	$\mathbb{Z}/2\mathbb{Z}$ when $u,v$ are both odd positive, 0 otherwise	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$(\mathbb{Z}/2\mathbb{Z})^{(p-1)/2}$ because that's the number of ordered pairs of positive odd numbers that add up to $p - 1$.	$(\mathbb{Z}/2\mathbb{Z})^{(p+3)/2}$
$p$ even positive	$\mathbb{Z}/2\mathbb{Z}$ for the cases $i,j$ both odd positive, 0 otherwise.	zero in all cases	$(\mathbb{Z}/2\mathbb{Z})^{p/2}$	0	$(\mathbb{Z}/2\mathbb{Z})^{p/2}$

Over an abelian group

The homology groups with coefficients in an abelian group $M$ are given as follows:

$H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/2M)^{(p+3)/2} \oplus (\operatorname{Ann}_M(2))^{(p-1)/2}, & \qquad p = 1,3,5,\dots\\ (M/2M)^{p/2} \oplus (\operatorname{Ann}_M(2))^{(p+2)/2}, & \qquad p = 2,4,6,\dots \\ M, & \qquad p = 0 \\\end{array}\right.$

Here, $M/2M$ is the quotient of $M$ by $2M = \{ 2x \mid x \in M \}$ and $\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}$.

These homology groups can be computed in terms of the homology groups over integers using the universal coefficients theorem for group homology.

Important case types for abelian groups

Case on $M$	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	$(M/2M)^{(p+3)/2}$	$(M/2M)^{p/2}$
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$	$(\operatorname{Ann}_M(2))^{(p-1)/2}$	$(\operatorname{Ann}_M(2))^{(p+2)/2}$
$M = \mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number	$(\mathbb{Z}/2\mathbb{Z})^{p+1}$	$(\mathbb{Z}/2\mathbb{Z})^{p+1}$
$M$ is a finite abelian group	isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$	isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$
$M$ is a finitely generated abelian group	all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$	all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + sp/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$

Cohomology groups for trivial group action

Over the integers

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

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

The first few cohomology groups are given below:

$p$	$\! 0$	$\! 1$	$\! 2$	$\! 3$	$\! 4$	$\! 5$
$H^p$	$\mathbb{Z}$	0	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$	$\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$
rank of $H^p$ as an elementary abelian 2-group	--	0	2	1	3	2

Over an abelian group

The cohomology groups with coefficients in an abelian group $M$ are given as follows:

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

Here, $M/2M$ is the quotient of $M$ by $2M = \{ 2x \mid x \in M \}$ and $\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}$.

These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.

The first few groups are given below:

$p$	$0$	$1$	$2$	$3$	$4$	$5$
$H^p$	$M$	$(\operatorname{Ann}_M(2))^2$	$(\operatorname{Ann}_M(2)) \oplus (M/2M)^2$	$(\operatorname{Ann}_M(2))^3 \oplus (M/2M)$	$(\operatorname{Ann}_M(2))^2 \oplus (M/2M)^3$	$(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2$

Important case types for abelian groups

Case on $M$	Conclusion about odd-indexed cohomology 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	$(M/2M)^{(p-3)/2}$	$(M/2M)^{(p+2)/2}$
$M$ is 2-divisible, but not necessarily uniquely so, e.g., $M = \mathbb{Q}/\mathbb{Z}$	$(\operatorname{Ann}_M(2))^{(p+3)/2}$	$(\operatorname{Ann}_M(2))^{p/2}$
$M = \mathbb{Z}/2^n\mathbb{Z}$, $n$ any natural number	$(\mathbb{Z}/2\mathbb{Z})^{p+1}$	$(\mathbb{Z}/2\mathbb{Z})^{p+1}$
$M$ is a finite abelian group	isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$	isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}$ where $r$ is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of $M$
$M$ is a finitely generated abelian group	all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p - 1)/2}$ where $r$ is the rank for the 2-Sylow subgroup of the torsion part of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$	all isomorphic to $(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of $M$

Cohomology ring

Over the integers

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Over a 2-divisible ring

If $R$ is a 2-divisible unital ring, then it is also a uniquely 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and $H^*(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};R)$ is isomorphic to $R$, occurring in the $H^0$.

In particular, this includes the case $R$ a field of characteristic not 2, as well as $R$ a ring (not necessarily a field) of finite positive characteristic.

Over characteristic two

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Second cohomology groups and extensions

Schur multiplier and Schur covering groups

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 isomorphic to cyclic group:Z2.

There are two possibilities for the Schur covering group: dihedral group:D8 and quaternion group. These belong to the Hall-Senior family $\Gamma_2$ (up to isoclinism). They are precisely the stem extensions where the acting group is the Klein four-group and the base group is its Schur multiplier, namely cyclic group:Z2. For more, see second cohomology group for trivial group action of V4 on Z2.

See also the projective representation theory of Klein four-group.

Second cohomology groups for trivial group action

Group acted upon	Order	Second part of GAP ID	Second cohomology group for trivial group action (as abstract group)	Order	Extensions	Number of extensions up to pseudo-congruence, i.e., number of orbits under action of relevant automorpism groups	Cohomology information
cyclic group:Z2	2	1	elementary abelian group:E8	8	elementary abelian group:E8, direct product of Z4 and Z2, quaternion group and dihedral group:D8	4	second cohomology group for trivial group action of V4 on Z2
cyclic group:Z4	4	1	elementary abelian group:E8	8	direct product of Z4 and V4, direct product of Z8 and Z2, central product of D8 and Z4, M16	4	second cohomology group for trivial group action of V4 on Z4
Klein four-group	4	2	elementary abelian group:E64	64	elementary abelian group:E16, direct product of Z4 and Z4, direct product of Z4 and V4, direct product of D8 and Z2, direct product of Q8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4	7	second cohomology group for trivial group action of V4 on V4
cyclic group:Z8	8	1	elementary abelian group:E8	8	direct product of Z8 and V4, direct product of Z16 and Z2, central product of D8 and Z8, M32	4	second cohomology group for trivial group action of V4 on Z8
direct product of Z4 and Z2	8	2	elementary abelian group:E64	64	?	?	second cohomology group for trivial group action of V4 on direct product of Z4 and Z2
elementary abelian group:E8	8	5	elementary abelian group:E512	512	?	?	second cohomology group for trivial group action of V4 on E8