Difference between revisions of "Group cohomology of dihedral group:D16"

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This article describes the homology and cohomology group of [[dihedral group:D16]], the dihedral group of order 16 and degree 8.
 
This article describes the homology and cohomology group of [[dihedral group:D16]], the dihedral group of order 16 and degree 8.
  
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==Family contexts==
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{| class="sortable" border="1"
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! Family name !! Parameter value !! Information on group cohomology of family
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|-
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| [[dihedral group]] <math>D_{2n}</math> of degree <math>n</math>, order <math>2n</math> || degree <math>n = 8</math>, order <math>2n = 16</math> || [[group cohomology of dihedral groups]]
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|}
 
==Homology groups for trivial group action==
 
==Homology groups for trivial group action==
  
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The first few homology groups are as follows:
 
The first few homology groups are as follows:
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{| class="sortable" border="1"
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! <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>
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|-
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| <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}/8\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}/8\mathbb{Z}</math> || <math>(\mathbb{Z}/2\mathbb{Z})^4</math>
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Latest revision as of 04:05, 16 January 2013

This article gives specific information, namely, group cohomology, about a particular group, namely: dihedral group:D16.
View group cohomology of particular groups | View other specific information about dihedral group:D16

This article describes the homology and cohomology group of dihedral group:D16, the dihedral group of order 16 and degree 8.

Family contexts

Family name Parameter value Information on group cohomology of family
dihedral group D_{2n} of degree n, order 2n degree n = 8, order 2n = 16 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 with coefficients in the integers are given as follows:

H_q(D_{16};\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}/8\mathbb{Z}, & q \equiv 3 \pmod 4 \\(\mathbb{Z}/2\mathbb{Z})^{q/2}, & q \mbox{ even }, q > 0 \\ \end{array}\right.

The first few homology groups are as follows:

q 0 1 2 3 4 5 6 7 8
H_q \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}/8\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} (\mathbb{Z}/2\mathbb{Z})^4 (\mathbb{Z}/2\mathbb{Z})^3 (\mathbb{Z}/2\mathbb{Z})^4 \oplus \mathbb{Z}/8\mathbb{Z} (\mathbb{Z}/2\mathbb{Z})^4