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Group cohomology of Klein four-group

11,187 bytes added, 03:38, 23 November 2012
Over an abelian group
<math>\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}</math>
* The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:
** The map <math>\mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i,j-1)}</math> is multiplication by zero if <math>j</math> is odd and is multiplication by two if <math>j</math> is even.
The homology groups with coefficients in the ring of integers <math>\mathbb{Z}</math> are given as follows:
<math>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.</math>
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"! <math>p</math> !! <math>\! 0 </math> !! <math>\! 1 </math> !! <math>\! 2</math> !! <math>\! 3 </math> !! <math>\! 4 </math> !! <math>\! 5</math>|-| <math>H_p</math> || <math>\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math>|-| rank of <math>H_p</math> 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 <math>G_1 = G_2 = \mathbb{Z}/2\mathbb{Z}</math> and <math>M = \mathbb{Z}</math> in the formula. {| class="sortable" border="1"! Case on <math>p</math> !! Value of <math>H_i(G_1;M) \otimes H_j(G_2;M)</math> where <math>i + j = p</math> !! Value of <math>\operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M))</math> where <math>u + v = p -1</math> !! Value of <math>\bigoplus_{i+j = p} H_i(G_1;M) \otimes H_j(G_2;M)</math> !! Value of <math>\bigoplus_{u + v = p - 1} \operatorname{Tor}_{\mathbb{Z}}^1(H_u(G_1;M),H_v(G_2;M))</math> !! Value of <math>H_p(G_1 \times G_2;M)</math> (sum of preceding two columns|-| <math>p = 0</math> || <math>\mathbb{Z}</math> in case <math>i = j = 0</math> || No such cases || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}</math>|-| <math>p</math> odd positive || <math>\mathbb{Z}/2\mathbb{Z}</math> for the case <math>i = 0, j = p</math> and the case <math>i = p, j = 0</math>. 0 in all other cases, because for <math>i + j</math> to be odd, at least one of <math>i</math> and <math>j</math> must be even, forcing the corresponding homology group to be 0. || <math>\mathbb{Z}/2\mathbb{Z}</math> when <math>u,v</math> are both odd positive, 0 otherwise || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^{(p-1)/2}</math> because that's the number of ordered pairs of positive odd numbers that add up to <math>p - 1</math>. || <math>(\mathbb{Z}/2\mathbb{Z})^{(p+3)/2}</math>|-| <math>p</math> even positive || <math>\mathbb{Z}/2\mathbb{Z}</math> for the cases <math>i,j</math> both odd positive, 0 otherwise. || zero in all cases || <math>(\mathbb{Z}/2\mathbb{Z})^{p/2}</math> || 0 || <math>(\mathbb{Z}/2\mathbb{Z})^{p/2}</math>|}</toggledisplay> ===Over an abelian group === The homology groups with coefficients in an abelian group <math>M</math> are given as follows: <math>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.</math> Here, <math>M/2M</math> is the quotient of <math>M</math> by <math>2M = \{ 2x \mid x \in M \}</math> and <math>\operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}</math>. 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 <math>M</math> !! Conclusion about odd-indexed homology groups, i.e., <math>H_p, p = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H_p, p = 2,4,6,\dots</math>|-| <math>M</math> is uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups|-| <math>M</math> is 2-torsion-free, i.e., no nonzero element of <math>M</math> doubles to zero || <math>(M/2M)^{(p+3)/2}</math> || <math>(M/2M)^{p/2}</math>|-| <math>M</math> is 2-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(2))^{(p-1)/2}</math> || <math>(\operatorname{Ann}_M(2))^{(p+2)/2}</math>|-| <math>M = \mathbb{Z}/2^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math>|-| <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math>|-| <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2}</math> where <math>r</math> is the rank for the 2-Sylow subgroup of the torsion part of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math> || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + sp/2}</math> where <math>r</math> is the rank for the 2-Sylow subgroup of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of <math>M</math>|} ==Cohomology groups for trivial group action== ===Over the integers=== The cohomology groups with coefficients in the integers are given as below: <math>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.</math>
The homology first few cohomology groups are given below: {| class="sortable" border="1"! <math>p</math> !! <math>\! 0 </math> !! <math>\! 1 </math> !! <math>\! 2</math> !! <math>\! 3 </math> !! <math>\! 4 </math> !! <math>\! 5</math>|-| <math>H^p</math> || <math>\mathbb{Z}</math> || 0 || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math>|-| rank of <math>H^p</math> as an elementary abelian 2-group || -- || 0 || 2 || 1 || 3 || 2|} ===Over an abelian group=== The cohomology groups with coefficients in an abelian group <math>M</math> are given as follows: <math>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.</math> Here, <math>M/2M</math> is the quotient of <math>M</math> by <math>2M = \{ 2x \mid x \in M \}</math> and <math>\operatorname{Ann}_M(which may 2) = \{ x \in M \mid 2x = 0 \}</math>. 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"! <math>p</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math>|-| <math>H^p</math> || <math>M</math> || <math>(\operatorname{Ann}_M(2))^2</math> || <math>(\operatorname{Ann}_M(2)) \oplus (M/2M)^2</math> || <math>(\operatorname{Ann}_M(2))^3 \oplus (M/2M)</math> || <math>(\operatorname{Ann}_M(2))^2 \oplus (M/2M)^3</math> || <math>(\operatorname{Ann}_M(2))^4 \oplus (M/2M)^2</math>|} ===Important case types for abelian groups=== {| class="sortable" border="1"! Case on <math>M</math> !! Conclusion about odd-indexed cohomology groups, i.e., <math>H^p, p = 1,3,5,\dots</math>!! Conclusion about even-indexed homology groups, i.e., <math>H^p, p = 2,4,6,\dots</math>|-| <math>M</math> is uniquely 2-divisible, i.e., every element of <math>M</math> has a unique half. This includes the case that <math>M</math> is a field of characteristic not 2. || all zero groups || all zero groups|-| <math>M</math> is 2-torsion-free, i.e., no nonzero element of <math>M</math> doubles to zero || <math>(M/2M)^{(p-3)/2}</math> || <math>(M/2M)^{(p+2)/2}</math>|-| <math>M</math> is 2-divisible, but not necessarily uniquely so, e.g., <math>M = \mathbb{Q}/\mathbb{Z}</math> || <math>(\operatorname{Ann}_M(2))^{(p+3)/2}</math> || <math>(\operatorname{Ann}_M(2))^{p/2}</math>|-| <math>M = \mathbb{Z}/2^n\mathbb{Z}</math>, <math>n</math> any natural number || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^{p+1}</math>|-| <math>M</math> is a finite abelian group || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math> || isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1)}</math> where <math>r</math> is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of <math>M</math>|-| <math>M</math> is a finitely generated abelian group || all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p - 1)/2}</math> where <math>r</math> is the rank for the 2-Sylow subgroup of the torsion part of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as a module over a ring free abelian group of the torsion-free part) of <math>RM</math>|| all isomorphic to <math>(\mathbb{Z}/2\mathbb{Z}) are given ^{r(p + 1) + s(p + 3)/2}</math> where <math>r</math> is the rank for the 2-Sylow subgroup of <math>M</math> and <math>s</math> is the free rank (i.e., the rank as follows:a free abelian group of the torsion-free part) of <math>M</math>|} ==Cohomology ring== ===Over the integers===
{{fillin}}
==Cohomology groups and cohomology =Over a 2-divisible ring=== If <math>R</math> 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 <math>H^*(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};R)</math> is isomorphic to <math>R</math>, occurring in the <math>H^0</math>.
===Groups over In particular, this includes the integers===case <math>R</math> a field of characteristic not 2, as well as <math>R</math> 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===
<math>H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Zfillin};\mathbb{Z}) = </math>
==Second cohomology groups and extensions==
===Schur multiplierand Schur covering groups=== The [[Schur multiplier]], defined as the [[second cohomology group for trivial group action]] <math>H^2(G,\mathbb{C}^\ast)</math> and also as the second homology group <math>H_2(G,\mathbb{Z})</math>, 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 <math>H^2(G,\mathbb{C}^\ast)Gamma_2</math> (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 <math>H_2(G,\mathbb{Z})</math>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]].
See also the [[projective representation theory of Klein four-group]].
===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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