This article gives specific information, namely, group cohomology, about a particular group, namely: Klein fourgroup.
View group cohomology of particular groups  View other specific information about Klein fourgroup
Classifying space and corresponding chain complex
The classifying space of the Klein fourgroup is the product space , where is infinitedimensional real projective space.
A chain complex that can be used to compute the homology of this space is
Homology groups
Over the integers
The homology groups with coefficients in the ring of integers are given as follows:
These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 using the Kunneth formula.
Over an abelian group
The homology groups with coefficients in an abelian group (which may be equipped with additional structure as a module over a ring ) are given as follows:
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Cohomology ring with coefficients in integers
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Second cohomology groups and extensions
Schur multiplier
The Schur multiplier, defined as the second cohomoogy group for trivial group action and also as the second homology group , is isomorphic to cyclic group:Z2.
See also the projective representation theory of Klein fourgroup.
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 
elementary abelian group:E8, direct product of Z4 and Z2, quaternion group and dihedral group:D8 
second cohomology group for trivial group action of V4 on Z2

cyclic group:Z4 
4 
1 
elementary abelian group:E8 
direct product of Z4 and V4, direct product of Z8 and Z2, central product of D8 and Z4, M16 
second cohomology group for trivial group action of V4 on Z4

Klein fourgroup 
4 
2 
elementary abelian group:E64 
elementary abelian group:E16, direct product of Z4 and Z4, direct product of Z4 and V4, direct product of D8 and Z2, direct product of Q8 and Z2, SmallGroup(16,3), nontrivial semidirect product of Z4 and Z4 
second cohomology group for trivial group action of V4 on V4

cyclic group:Z8 
8 
1 
elementary abelian group:E8 
direct product of Z8 and V4, direct product of Z16 and Z2, central product of D8 and Z8, M32 
second cohomology group for trivial group action of V4 on Z8

direct product of Z4 and Z2 
8 
2 
? 
? 
?

elementary abelian group:E8 
8 
5 
? 
? 
?
