Group cohomology of dihedral group:D16: Difference between revisions

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:

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

@@ Line 6: / Line 6: @@
 This article describes the homology and cohomology group of [[dihedral group:D16]], the dihedral group of order 16 and degree 8.
+==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 = 8</math>, order <math>2n = 16</math> || [[group cohomology of dihedral groups]]
+|}
 ==Homology groups for trivial group action==
@@ Line 17: / Line 24: @@
 The first few homology groups are as follows:
+{| 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}/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>
+|}