Group cohomology of dihedral groups: Difference between revisions
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The homology groups with coefficients in an abelian group <math>M</math> are as follows when the degree <math>n</math> is odd: | The homology groups with coefficients in an abelian group <math>M</math> are as follows when the degree <math>n</math> is odd: | ||
<math>H_p(D_{2n};M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann} | <math>H_p(D_{2n};M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann}_M(2) & \qquad p \equiv 2 \pmod 4 \\ M/2nM, & \qquad p \equiv 3 \pmod 4\\ \operatorname{Ann}_M(2n), & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.</math> | ||
Here, <math>\operatorname{Ann} | Here, <math>\operatorname{Ann}_M(2)</math> denotes the 2-torsion subgroup of <math>M</math> and <math>\operatorname{Ann}_M(2n)</math> denotes the <math>2n</math>-torsion subgroup of <math>M</math>. | ||
==Cohomology groups for trivial group action== | ==Cohomology groups for trivial group action== | ||
Revision as of 02:24, 12 October 2011
This article gives specific information, namely, group cohomology, about a family of groups, namely: group cohomology.
View group cohomology of group families | View other specific information about group cohomology
We consider here the dihedral group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle D_{2n}} of order Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} and degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} (i.e., its natural action is on a set of size Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} ).
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 for odd degree
The homology groups with coefficients in the ring of integers are as follows when the degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \! H_p(D_{2n};\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}/2n\mathbb{Z}, & \qquad p \equiv 3 \pmod 4 \\ 0, & \qquad p \ne 0, p \ \operatorname{even}\\\end{array}\right.}
Over an abelian group for odd degree
The homology groups with coefficients in an abelian group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} are as follows when the degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H_p(D_{2n};M) = \left \lbrace \begin{array}{rl} M, & \qquad p = 0 \\ M/2M, & \qquad p \equiv 1 \pmod 4 \\ \operatorname{Ann}_M(2) & \qquad p \equiv 2 \pmod 4 \\ M/2nM, & \qquad p \equiv 3 \pmod 4\\ \operatorname{Ann}_M(2n), & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.}
Here, Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ann}_M(2)} denotes the 2-torsion subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ann}_M(2n)} denotes the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} -torsion subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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 for odd degree
The cohomology groups with coefficients in the ring of integers are as follows when the degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^p(D_{2n};\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}/2n\mathbb{Z}, & \qquad p \ne 0, p \equiv 0 \pmod 4\\ 0, & \qquad p \ \operatorname{odd} \\\end{array}\right.}
Over an abelian group for odd degree
The cohomology groups with coefficients in an abelian group Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} are as follows when the degree Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n} is odd:
Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle H^p(D_{2n};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}_{2n}(M), & \qquad p \equiv 3 \pmod 4\\ M/2nM, & \qquad p > 0, p \equiv 0 \pmod 4 \\\end{array}\right.}
Here Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ann}_2(M)} denotes the 2-torsion subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \operatorname{Ann}_{2n}(M)} denotes the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2n} -torsion subgroup of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle M} .