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Group cohomology of Klein four-group

4,233 bytes added, 03:38, 23 November 2012
Over an abelian group
{| class="sortable" border="1"
! Case on <math>R</math> or <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
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(2) = \{ x \in M \mid 2x = 0 \}</math>.
These can be deduced from the homology groups with coefficients in the integers using the [[dual universal coefficients theoremfor 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 free abelian group of the torsion-free part) of <math>M</math> || 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 <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 ring==
 
===Over the integers===
{{fillin}}
==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>. In particular, this includes the 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. ===Over characteristic two=== {{fillin}}
==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]].
 
There are two possibilities for the [[Schur covering group]]: [[dihedral group:D8]] and [[quaternion group]]. These belong to the Hall-Senior family <math>\Gamma_2</math> (up to [[isoclinism]]). They are precisely the [[stem extension]]s 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===
<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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