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

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(Over an abelian group)
(Over the integers)
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The cohomology groups with coefficients in the ring of integers are as follows:
 
The cohomology groups with coefficients in the ring of integers are as follows:
  
<math>H^p(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 2 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & p \ \operatorname{odd} \\\end{array}\right.</math>
+
<math>H^p(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 2 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & \qquad p \ \operatorname{odd} \\\end{array}\right.</math>
  
 
===Over an abelian group===
 
===Over an abelian group===

Revision as of 01:45, 9 October 2011

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

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 ring of integers are as follows:

\! H_p(D_8;\mathbb{Z}) = \left\lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 1 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even}\\\end{array}\right.

As a sequence (Starting p = 0), the first few homology groups are:

p 0 1 2 3 4 5 6 7 8
H^p(D_8;\mathbb{Z}) \mathbb{Z} \mathbb{Z}/2\mathbb{Z} 0 \mathbb{Z}/8\mathbb{Z} 0 \mathbb{Z}/2\mathbb{Z} 0 \mathbb{Z}/8\mathbb{Z} 0

Over an abelian group

Over an abelian group

The homology groups with coefficients in an abelian group M are as follows:

H_p(D_8;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann}_2(M) & \qquad p \equiv 2 \pmod 4 \\ M/8M, & \qquad p \equiv 3 \pmod 4\\ \operatorname{Ann}_8(M), & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.

Here, \operatorname{Ann}_2(M) denotes the 2-torsion subgroup of M and \operatorname{Ann}_8(M) denotes the 8-torsion subgroup of M.

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 with coefficients in the ring of integers are as follows:

H^p(D_8;\mathbb{Z}) = \left \lbrace \begin{array}{rl} \mathbb{Z}, & \qquad p = 0 \\ \mathbb{Z}/2\mathbb{Z}, & \qquad p \equiv 2 \pmod 4 \\ \mathbb{Z}/8\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & \qquad p \ \operatorname{odd} \\\end{array}\right.

Over an abelian group

The cohomology groups with coefficients in an abelian group M are as follows:

H^p(D_8;M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ \operatorname{Ann}_2(M), & \qquad p \equiv 1 \pmod 4 \\ M/2M & \qquad p \equiv 2 \pmod 4 \\ \operatorname{Ann}_8(M), & \qquad p \equiv 3 \pmod 4\\ M/8M, & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.

Here \operatorname{Ann}_2(M) denotes the 2-torsion subgroup of M and \operatorname{Ann}_8(M) denotes the 8-torsion subgroup of M.

Cohomology ring with coefficients in integers

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

Second cohomology groups and extensions

Second cohomology groups for trivial group action

Group acted upon Order Second part of GAP ID Second cohomology group for trivial group action Extensions Cohomology information
cyclic group:Z2 2 1 elementary abelian group:E8 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 second cohomology group for trivial group action of D8 on Z2
cyclic group:Z4 4 1  ?  ? second cohomology group for trivial group action of D8 on Z4