Group cohomology of Klein four-group
This article gives specific information, namely, group cohomology, about a particular group, namely: Klein four-group.
View group cohomology of particular groups | View other specific information about Klein four-group
Classifying space and corresponding chain complex
The classifying space of the Klein four-group is the product space , where is infinite-dimensional real projective space.
A chain complex that can be used to compute the homology of this space is given as follows:
- The chain group is a sum of copies of , indexed by ordered pairs where . In other words, the chain group is:
- The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:
- The map is multiplication by zero if is odd and is multiplication by two if is even.
- The map is multiplication by zero if is odd and multiplication by two if is even.
Over the integers
The homology groups with coefficients in the ring of integers are given as follows:
The first few homology groups are given below:
|rank of as an elementary abelian 2-group||--||2||1||3||2||4|
These homology groups can be obtained from the knowledge of the homology groups of cyclic group:Z2 (see group cohomology of cyclic group:Z2) using the Kunneth formula for group homology. They can also be computed explicitly using the chain complex description above.Here is the computation using the Kunneth formula for group homology: [SHOW MORE]
Over an abelian group
The homology groups with coefficients in an abelian group are given as follows:
Here, is the quotient of by and .
These homology groups can be computed in terms of the homology groups over integers using the universal coefficients theorem for group homology.
Important case types for abelian groups
|Case on or||Conclusion about odd-indexed homology groups, i.e.,||Conclusion about even-indexed homology groups, i.e.,|
|is uniquely 2-divisible, i.e., every element of has a unique half. This includes the case that is a field of characteristic not 2.||all zero groups||all zero groups|
|is 2-torsion-free, i.e., no nonzero element of doubles to zero|
|is 2-divisible, but not necessarily uniquely so, e.g.,|
|, any natural number|
|is a finite abelian group||isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of||isomorphic to where is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of|
|is a finitely generated abelian group||all isomorphic to where is the rank for the 2-Sylow subgroup of the torsion part of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of||all isomorphic to where is the rank for the 2-Sylow subgroup of and is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of|
Cohomology groups for trivial group action
Over the integers
The cohomology groups with coefficients in the integers are given as below:
The first few cohomology groups are given below:
|rank of as an elementary abelian 2-group||--||0||2||1||3||2|
Second cohomology groups and extensions
See also the projective representation theory of Klein four-group.