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.
- The map
Homology groups
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:
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rank of ![]() |
-- | 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 ![]() |
Conclusion about odd-indexed homology groups, i.e., ![]() |
Conclusion about even-indexed homology groups, i.e., ![]() |
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all zero groups | all zero groups |
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isomorphic to ![]() ![]() ![]() |
isomorphic to ![]() ![]() ![]() |
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all isomorphic to ![]() ![]() ![]() ![]() ![]() |
all isomorphic to ![]() ![]() ![]() ![]() ![]() |
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:
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---|---|---|---|---|---|---|
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0 | ![]() |
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rank of ![]() |
-- | 0 | 2 | 1 | 3 | 2 |
Over an abelian group
The cohomology groups with coefficients in an abelian group are given as follows:
Here, is the quotient of
by
and
.
These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.
The first few groups are given below:
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---|---|---|---|---|---|---|
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Important case types for abelian groups
Case on ![]() |
Conclusion about odd-indexed cohomology groups, i.e., ![]() |
Conclusion about even-indexed homology groups, i.e., ![]() |
---|---|---|
![]() ![]() ![]() |
all zero groups | all zero groups |
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isomorphic to ![]() ![]() ![]() |
isomorphic to ![]() ![]() ![]() |
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all isomorphic to ![]() ![]() ![]() ![]() ![]() |
all isomorphic to ![]() ![]() ![]() ![]() ![]() |
Cohomology ring
Over the integers
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Over a 2-divisible ring
If is a 2-divisible unital ring, then it is also a uniquely 2-divisible ring. In this case, all cohomology groups in positive degrees vanish, and
is isomorphic to
, occurring in the
.
In particular, this includes the case a field of characteristic not 2, as well as
a ring (not necessarily a field) of finite positive characteristic.
Over characteristic two
PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]Second cohomology groups and extensions
Schur multiplier and Schur covering groups
The Schur multiplier, defined as the second cohomology group for trivial group action and also as the second homology group
, is isomorphic to cyclic group:Z2.
There are two possibilities for the Schur covering group: dihedral group:D8 and quaternion group. These belong to the Hall-Senior family (up to isoclinism). They are precisely the stem extensions where the acting group is the Klein four-group and the base group is its Schur multiplier, namely cyclic group:Z2. For more, see second cohomology group for trivial group action of V4 on Z2.
See also the projective representation theory of Klein four-group.