# Difference between revisions of "Group cohomology of dihedral group:D8"

(→Cohomology groups for trivial group action) |
(→Baer invariants) |
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group = dihedral group:D8| | group = dihedral group:D8| | ||

connective = of}} | 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== | ==Homology groups for trivial group action== | ||

Line 10: | Line 18: | ||

===Over the integers=== | ===Over the integers=== | ||

− | The homology groups | + | The homology groups over the integers are given as follows: |

− | <math> | + | <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=" | + | {| 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> | + | | <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=== | ===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== | ==Cohomology groups for trivial group action== | ||

Line 35: | Line 49: | ||

===Over the integers=== | ===Over the integers=== | ||

− | The cohomology groups | + | The cohomology groups over the integers are given as follows: |

− | <math>H^ | + | <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=== | ===Over an abelian group=== | ||

− | The cohomology groups | + | The cohomology groups over an abelian group <math>M</math> are given as follows: |

− | <math>H^ | + | <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== | ==Cohomology ring with coefficients in integers== | ||

Line 52: | Line 80: | ||

==Second cohomology groups and extensions== | ==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=== | ===Second cohomology groups for trivial group action=== | ||

{| class="sortable" border="1" | {| class="sortable" border="1" | ||

− | ! Group acted upon !! Order !! Second part of GAP ID !! [[Second cohomology group for trivial group action]] !! Extensions !! Cohomology information | + | ! 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> |

## Latest revision as of 00:27, 29 May 2013

This article gives specific information, namely, group cohomology, about a particular group, namely: dihedral group:D8.

View group cohomology of particular groups | View other specific information about dihedral group:D8

## Family contexts

Family name | Parameter value | Information on group cohomology of family |
---|---|---|

dihedral group of degree , order | degree , order | group cohomology of dihedral groups |

## Homology groups for trivial group action

FACTS TO CHECK AGAINST(homology group for trivial group action):

First homology group: first homology group for trivial group action equals tensor product with abelianization

Second homology group: formula for second homology group for trivial group action in terms of Schur multiplier and abelianization|Hopf's formula for Schur multiplier

General: universal coefficients theorem for group homology|homology group for trivial group action commutes with direct product in second coordinate|Kunneth formula for group homology

### Over the integers

The homology groups over the integers are given as follows:

The first few homology groups are given below:

### Over an abelian group

The homology groups over an abelian group are given as follows:

The first few homology groups with coefficients in an abelian group are given below:

## Cohomology groups for trivial group action

FACTS TO CHECK AGAINST(cohomology group for trivial group action):

First cohomology group: first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms

Second cohomology group: formula for second cohomology group for trivial group action in terms of Schur multiplier and abelianization

In general: dual universal coefficients theorem for group cohomology relating cohomology with arbitrary coefficientsto homology with coefficients in the integers. |Cohomology group for trivial group action commutes with direct product in second coordinate | Kunneth formula for group cohomology

### Over the integers

The cohomology groups over the integers are given as follows:

The first few cohomology groups are given below:

0 |

### Over an abelian group

The cohomology groups over an abelian group are given as follows:

The first few cohomology groups with coefficients in an abelian group are:

## Cohomology ring with coefficients in integers

**PLACEHOLDER FOR INFORMATION TO BE FILLED IN**: [SHOW MORE]

## Second cohomology groups and extensions

### Schur multiplier

The Schur multiplier, defined as second cohomology group for trivial group action , and also as the second homology group , is cyclic group:Z2.

This has implications for projective representation theory of dihedral group:D8.

### Schur covering groups

The three possible Schur covering groups 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 extensions.

### Second cohomology groups for trivial group action

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-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 | [SHOW MORE] | 11 | second cohomology group for trivial group action of D8 on V4 |

### Baer invariants

Subvariety of the variety of groups | General name of Baer invariant | Value of Baer invariant for this group |
---|---|---|

abelian groups | Schur multiplier | cyclic group:Z2 |

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 can be computed in GAP using the GroupHomology function in the `HAP` package, which can be loaded by the command `LoadPackage("hap");` 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

gap> GroupHomology(DihedralGroup(8),1); [ 2, 2 ]

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 is , which is the Klein four-group.

#### Computation of second homology group

gap> GroupHomology(DihedralGroup(8),2); [ 2 ]

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:

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 ] ] ]