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Group cohomology of Klein four-group

, 03:38, 23 November 2012
Over an abelian group
$\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}$
* 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 homology groups with coefficients in the ring of integers $\mathbb{Z}$ are given as follows:
$H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(p + 13)/2}, & p = 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{p/2}, & p = 2,4,6,\dots \\ \mathbb{Z}, & p = 0 \\\end{array}\right.$
These The first few 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 [[topospacesare given below:Kunneth formula|Kunneth formula]]. They can also be computed explicitly using the chain complex description above.
{| class="sortable" border="1"! $p$ !! $\! 0$ !! $\! 1$ !! $\! 2$ !! $\! 3$ !! $\! 4$ !! $\! 5$|-| $H_p$ || $\mathbb{Z}$ || $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ || $\mathbb{Z}/2\mathbb{Z}$ || $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ || $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$ || $\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}$|-| 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: <toggledisplay> We set $G_1 = G_2 = \mathbb{Z}/2\mathbb{Z}$ and $M = \mathbb{Z}$ in the formula. {| class="sortable" border="1"! 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 || $(\mathbb{Z}/2\mathbb{Z})^{p/2}$|}</toggledisplay> ===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=== {| class="sortable" border="1"! 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 homology first few cohomology groups are given below: {| class="sortable" border="1"! $p$ !! $\! 0$ !! $\! 1$ !! $\! 2$ !! $\! 3$ !! $\! 4$ !! $\! 5$|-| $H^p$ || $\mathbb{Z}$ || 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 $\operatorname{Ann}_M(which may 2) = \{ x \in M \mid 2x = 0 \}$. These can be equipped deduced from the homology groups with additional structure coefficients in the integers using the [[dual universal coefficients theorem for group cohomology]]. The first few groups are given below: {| class="sortable" border="1"! $p$ !! $0$ !! $1$ !! $2$ !! $3$ !! $4$ !! $5$|-| $H^p$ || $M$ || $(\operatorname{Ann}_M(2))^2$ || $(\operatorname{Ann}_M(2)) \oplus (M/2M)^2$ || $(\operatorname{Ann}_M(2))^3 \oplus (M/2M)$ || $(\operatorname{Ann}_M(2))^2 \oplus (M/2M)^3$ || $(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2$|} ===Important case types for abelian groups=== {| class="sortable" border="1"! 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 module over a ring free abelian group of the torsion-free part) of $RM$|| all isomorphic to $(\mathbb{Z}/2\mathbb{Z}) are given ^{r(p + 1) + s(p + 3)/2}$ where $r$ is the rank for the 2-Sylow subgroup of $M$ and $s$ is the free rank (i.e., the rank as follows:a free abelian group of the torsion-free part) of $M$|} ==Cohomology ring== ===Over the integers===
{{fillin}}
==Cohomology groups and cohomology =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 $H^0$.
===Groups over In particular, this includes the integers===case $R$ a field of characteristic not 2, as well as $R$ a ring (not necessarily a field) of finite positive characteristic.
The cohomology groups with coefficients in the integers are given as below:===Over characteristic two===
$H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Zfillin};\mathbb{Z}) =$
==Second cohomology groups and extensions==
===Schur multiplierand 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]].
The There are two possibilities for the [[Schur multipliercovering group]], defined as the : [[second cohomoogy dihedral group for trivial :D8]] and [[quaternion group action]] . These belong to the Hall-Senior family $H^2(G,\mathbb{C}^\ast)Gamma_2$ (up to [[isoclinism]]). They are precisely the [[stem extension]]s where the acting group is the [[Klein four-group]] and also as the second homology base group $H_2(G,\mathbb{Z})$is its Schur multiplier, is isomorphic to namely [[cyclic group:Z2]]. For more, see [[second cohomology group for trivial group action of V4 on Z2]].
===Second cohomology groups for trivial group action===
<section begin="second cohomology group for trivial group action summary"/>
{| class="sortable" border="1"
! 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-congruent extension|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]]
|}
<section end="second cohomology group for trivial group action summary"/>
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