Group cohomology of quaternion group
From Groupprops
This article gives specific information, namely, group cohomology, about a particular group, namely: quaternion group.
View group cohomology of particular groups | View other specific information about quaternion group
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 are given as follows:
The group is a finite group with periodic cohomology, in keeping with the other definition of being a group with periodic cohomology: every abelian subgroup is cyclic.
The first few homology groups are given below:
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Over an abelian group
The first few homology groups with coefficients in an abelian group are given below:
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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
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? | ? | ? | ? |
Over an abelian group
The first few cohomology groups with coefficients in an abelian group are as follows:
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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 the trivial group.
Schur covering groups
Since the Schur multiplier is a trivial group, the Schur covering group of the quaternion group is the quaternion group itself.
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 |
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cyclic group:Z2 | 2 | 1 | Klein four-group | direct product of Q8 and Z2 (1 time), nontrivial semidirect product of Z4 and Z4 (3 times) | second cohomology group for trivial group action of Q8 on Z2 |
cyclic group:Z4 | 4 | 1 | Klein four-group | direct product of Q8 and Z4 (1 time), nontrivial semidirect product of Z4 and Z8 (3 times) | second cohomology group for trivial group action of Q8 on Z4 |
Klein four-group | 4 | 2 | elementary abelian group:E16 | direct product of Q8 and V4 (1 time), SmallGroup(32,2), direct product of SmallGroup(16,4) and Z2 | second cohomology group for trivial group action of Q8 on V4 |