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Group cohomology of dihedral group:D8

9,448 bytes added, 00:27, 29 May 2013
Baer invariants
group = dihedral group:D8|
connective = of}}
 
==Family contexts==
 
{| class="sortable" border="1"
! Family name !! Parameter value !! Information on group cohomology of family
|-
| [[dihedral group]] <math>D_{2n}</math> of degree <math>n</math>, order <math>2n</math> || degree <math>n = 4</math>, order <math>2n = 8</math> || [[group cohomology of dihedral groups]]
|}
==Homology groups for trivial group action==
===Over the integers===
 
The homology groups over the integers are given as follows:
 
<math>H_q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & q = 0 \\(\mathbb{Z}/2\mathbb{Z})^{(q + 3)/2}, & q \equiv 1 \pmod 4\\ (\mathbb{Z}/2\mathbb{Z})^{(q + 1)/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 3 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \mbox{ even }, q > 0 \\ \end{array}\right.</math>
 
The first few homology groups are given below:
 
{| class="sortable" border="1"
! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> !! <math>8</math>
|-
| <math>H_q</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}/4\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4</math> || <math>(\mathbb{Z}/2\mathbb{Z})^3</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4 \oplus \mathbb{Z}/4\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4</math>
|}
===Over an abelian group===
 
The homology groups over an abelian group <math>M</math> are given as follows:
 
<math>H_q(D_8;M) = \left \lbrace \begin{array}{rl} M, & q = 0 \\(M/2M)^{(q + 3)/2} \oplus (\operatorname{Ann}_M(2))^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (M/2M)^{q/2} \oplus (\operatorname{Ann}_M(2))^{(q + 2)/2}, & q \equiv 2 \pmod 4 \\(M/2M)^{(q + 1)/2} \oplus M/4M \oplus (\operatorname{Ann}_M(2))^{(q - 1)/2}, & q \equiv 3 \pmod 4 \\(M/2M)^{q/2} \oplus (\operatorname{Ann}_M(2))^{q/2} \oplus \operatorname{Ann}_M(4), & q \equiv 0 \pmod 4, q > 0 \\ \end{array}\right.</math>
The first few homology groups with coefficients in an abelian group <math>M</math> are given below:
 
{| class="sortable" border="1"
! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> !! <math>8</math>
|-
| <math>H_q</math> || <math>M</math> || <math>M/2M \oplus M/2M</math> || <math>M/2M \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)</math> || <math>(M/2M)^2 \oplus M/4M \oplus \operatorname{Ann}_M(2)</math> || <math>(M/2M)^2 \oplus (\operatorname{Ann}_M(2))^2 \oplus \operatorname{Ann}_M(4)</math> || <math>(M/2M)^4 \oplus (\operatorname{Ann}_M(2))^2</math> || <math>(M/2M)^3 \oplus (\operatorname{Ann}_M(2))^4</math> || <math>(M/2M)^4 \oplus M/4M \oplus (\operatorname{Ann}_M(2))^3</math> || <math>(M/2M)^4 \oplus (\operatorname{Ann}_M(2))^4 \oplus \operatorname{Ann}_M(4)</math>
|}
==Cohomology groups for trivial group action==
===Over the integers===
 
The cohomology groups over the integers are given as follows:
 
<math>H^q(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & q =0 \\(\mathbb{Z}/2\mathbb{Z})^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (\mathbb{Z}/2\mathbb{Z})^{(q/2} \oplus \mathbb{Z}/4\mathbb{Z}, & q \equiv 0 \pmod 4, q \ne 0\\(\mathbb{Z}/2\mathbb{Z})^{(q+2)/2}, & q \equiv 2 \pmod 4 \\ \end{array}\right.</math>
 
The first few cohomology groups are given below:
 
{| class="sortable" border="1"
! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math> !! <math>8</math> !! <math>9</math>
|-
| <math>H^q</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}/4\mathbb{Z}</math> || <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4</math> || <math>(\mathbb{Z}/2\mathbb{Z})^3</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4 \oplus \mathbb{Z}/4\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4</math>
|}
===Over an abelian group===
 
The cohomology groups over an abelian group <math>M</math> are given as follows:
 
<math>H^q(D_8;M) = \left \lbrace \begin{array}{rl} M, & q = 0 \\(\operatorname{Ann}_M(2))^{(q + 3)/2} \oplus (M/2M)^{(q - 1)/2}, & q \equiv 1 \pmod 4\\ (\operatorname{Ann}_M(2))^{q/2} \oplus (M/2M)^{(q + 2)/2}, & q \equiv 2 \pmod 4 \\(\operatorname{Ann}_M(2))^{(q + 1)/2} \oplus \operatorname{Ann}_M(4) \oplus (M/2M)^{(q - 1)/2}, & q \equiv 3 \pmod 4 \\(\operatorname{Ann}_M(2))^{q/2} \oplus (M/2M)^{q/2} \oplus M/4M, & q \equiv 0 \pmod 4, q > 0 \\ \end{array}\right.</math>
 
The first few cohomology groups with coefficients in an abelian group <math>M</math> are:
 
{| class="sortable" border="1"
! <math>q</math> !! <math>0</math> !! <math>1</math> !! <math>2</math> !! <math>3</math> !! <math>4</math> !! <math>5</math> !! <math>6</math> !! <math>7</math>
|-
| <math>H^q</math> || <math>M</math> || <math>\operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2)</math> || <math>M/2M \oplus M/2M \oplus \operatorname{Ann}_M(2)</math> || <math>M/2M \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(2) \oplus \operatorname{Ann}_M(4)</math> || <math>(M/2M)^2 \oplus M/4M \oplus (\operatorname{Ann}_M(2))^2</math> || <math>(M/2M)^2 \oplus (\operatorname{Ann}_M(2))^4</math> || <math>(M/2M)^4 \oplus (\operatorname{Ann}_M(2))^3</math> || <math>(M/2M)^3 \oplus (\operatorname{Ann}_M(2))^4 \oplus \operatorname{Ann}_M(4)</math>
|}
==Cohomology ring with coefficients in integers==
==Second cohomology groups and extensions==
 
===Schur multiplier===
 
The [[Schur multiplier]], defined as [[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 [[cyclic group:Z2]].
 
This has implications for [[projective representation theory of dihedral group:D8]].
 
===Schur covering groups===
 
The three possible [[Schur covering group]]s for [[dihedral group:D8]] are: [[dihedral group:D16]], [[semidihedral group:SD16]], and [[generalized quaternion group:Q16]]. For more, see [[second cohomology group for trivial group action of D8 on Z2]], where these correspond precisely to the [[stem extension]]s.
===Second cohomology groups for trivial group action===
{| class="sortable" border="1"
! Group acted upon !! Order !! Second part of GAP ID !! [[Second cohomology group for trivial group action]] (as an abstract group) !! Order of second cohomology group !! Extensions !! Number of extensions up to [[pseudo-congruent extensions|pseudo-congruence]], i.e., number or orbits under automorphism group actions !! Cohomology information
|-
| [[cyclic group:Z2]] || 2 || 1 || [[elementary abelian group:E8]] || 8 || [[direct product of D8 and Z2]], [[SmallGroup(16,3)]], [[nontrivial semidirect product of Z4 and Z4]], [[dihedral group:D16]], [[semidihedral group:SD16]], [[generalized quaternion group:Q16]] || 6 || [[second cohomology group for trivial group action of D8 on Z2]]
|-
| [[cyclic group:Z4]] || 4 || 1 || ? [[elementary abelian group:E8]] || ? 8 || [[direct product of D8 and Z4]], [[nontrivial semidirect product of Z4 and Z8]], [[SmallGroup(32,5)]], [[central product of D16 and Z4]], [[SmallGroup(32,15)]], [[wreath product of Z4 and Z2]] || 6 || [[second cohomology group for trivial group action of D8 on Z4]]|-| [[Klein four-group]] || 4 || 2 || [[elementary abelian group:E64]] || 64 || <toggledisplay>[[direct product of D8 and V4]], [[direct product of SmallGroup(16,3) and Z2]], [[direct product of SmallGroup(16,4) and Z2]], [[SmallGroup(32,2)]], [[SmallGroup(32,9)]], [[SmallGroup(32,10)]], [[semidirect product of Z8 and Z4 of semidihedral type]], [[semidirect product of Z8 and Z4 of dihedral type]], [[direct product of D16 and Z2]], [[direct product of SD16 and Z2]], [[direct product of Q16 and Z2]]</toggledisplay> || 11 || [[second cohomology group for trivial group action of D8 on V4]]
|}
 
===Baer invariants===
 
{| class="sortable" border="1"
! Subvariety of the variety of groups !! General name of [[Baer invariant]] !! Value of Baer invariant for this group
|-
| [[abelian group]]s || [[Schur multiplier]] || [[cyclic group:Z2]]
|-
| [[group of nilpotency class two|groups of nilpotency class at most two]] || 2-[[nilpotent multiplier]] ||
|-
| groups of nilpotency class at most three || 3-[[nilpotent multiplier]] ||
|-
| any variety of groups containing all groups of nilpotency class at most three || -- ||
|}
 
==GAP implementation==
 
===Computation of integral homology===
 
The homology groups for trivial group action with coefficients in <math>\mathbb{Z}</math> can be computed in GAP using the [[GAP:GroupHomology|GroupHomology]] function in the <tt>HAP</tt> package, which can be loaded by the command <tt>LoadPackage("hap");</tt> if it is installed but not loaded. The function outputs the orders of cyclic groups for which the homology or cohomology group is the direct product of these (more technically, it outputs the elementary divisors for the homology or cohomology group that we are trying to compute).
 
Here are computations of the first few homology groups:
 
====Computation of first homology group====
 
<pre>gap> GroupHomology(DihedralGroup(8),1);
[ 2, 2 ]</pre>
 
The way this is to be interpreted is that the first homology group (the [[abelianization]]) is the direct sum of cyclic groups of the orders listed, so in this case we get that <math>H_1(D_8;\mathbb{Z})</math> is <math>\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}</math>, which is the [[Klein four-group]].
 
====Computation of second homology group====
<pre>gap> GroupHomology(DihedralGroup(8),2);
[ 2 ]</pre>
 
This says that the second homology group (the [[Schur multiplier]]) is [[cyclic group:Z2]].
 
====Computation of first few homology groups====
 
To compute the first eight homology groups, do:
 
<pre>gap> List([1,2,3,4,5,6,7,8],i->[i,GroupHomology(DihedralGroup(8),i)]);
[ [ 1, [ 2, 2 ] ], [ 2, [ 2 ] ], [ 3, [ 2, 2, 4 ] ], [ 4, [ 2, 2 ] ], [ 5, [ 2, 2, 2, 2 ] ], [ 6, [ 2, 2, 2 ] ], [ 7, [ 2, 2, 2, 2, 4 ] ], [ 8, [ 2, 2, 2, 2 ] ] ]</pre>
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