Difference between revisions of "Group cohomology of Klein four-group"

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The first few homomology groups are given below:
 
The first few homomology groups are given below:
  
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! <math>p</math> !! <math>\! 0 </math> !! <math>\! 1 </math> !! <math>\! 2</math> !! <math>\! 3 </math> !! <math>\! 4 </math> !! <math>\! 5</math>
 
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Revision as of 22:44, 12 October 2011

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 \mathbb{R}\mathbb{P}^\infty \times \mathbb{R}\mathbb{P}^\infty, where \mathbb{R}\mathbb{P}^\infty 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 n^{th} chain group is a sum of n + 1 copies of \mathbb{Z}, indexed by ordered pairs (i,j) where i + j = n. In other words, the n^{th} chain group is:

\mathbb{Z}_{(0,n)} \oplus \mathbb{Z}_{(1,n-1)} \oplus \dots \mathbb{Z}_{(n,0)}

  • The boundary map is (up to some sign issues that need to be fixed!) given by adding up the following maps:
    • The map \mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i,j-1)} is multiplication by zero if j is odd and is multiplication by two if j is even.
    • The map \mathbb{Z}_{(i,j)} \to \mathbb{Z}_{(i-1,j)} is multiplication by zero if i is odd and multiplication by two if i is even.

Homology groups

Over the integers

The homology groups with coefficients in the ring of integers \mathbb{Z} are given as follows:

H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = \left\lbrace\begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(p + 3)/2}, & p = 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{p/2}, & p = 2,4,6,\dots \\ \mathbb{Z}, & p = 0 \\\end{array}\right.

The first few homomology groups are given below:

p \! 0 \! 1 \! 2 \! 3 \! 4 \! 5
H_p \mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z}
rank of H_p 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 M are given as follows:

H_p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/2M)^{(p+3)/2} \oplus (\operatorname{Ann}_M(2))^{(p-1)/2}, & \qquad p = 1,3,5,\dots\\ (M/2M)^{p/2} \oplus (\operatorname{Ann}_M(2))^{(p+2)/2}, & \qquad p = 2,4,6,\dots \\ M, & \qquad p = 0 \\\end{array}\right.

Here, M/2M is the quotient of M by 2M = \{ 2x \mid x \in M \} and \operatorname{Ann}_M(2) = \{ x \in M \mid 2x = 0 \}.

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 R or M Conclusion about odd-indexed homology groups, i.e., H_p, p = 1,3,5,\dots Conclusion about even-indexed homology groups, i.e., H_p, p = 2,4,6,\dots
M is uniquely 2-divisible, i.e., every element of M has a unique half. This includes the case that M is a field of characteristic not 2. all zero groups all zero groups
M is 2-torsion-free, i.e., no nonzero element of M doubles to zero (M/2M)^{(p+3)/2} (M/2M)^{p/2}
M is 2-divisible, but not necessarily uniquely so, e.g., M = \mathbb{Q}/\mathbb{Z} (\operatorname{Ann}_M(2))^{(p-1)/2} (\operatorname{Ann}_M(2))^{(p+2)/2}
M = \mathbb{Z}/2^n\mathbb{Z}, n any natural number (\mathbb{Z}/2\mathbb{Z})^{p+1} (\mathbb{Z}/2\mathbb{Z})^{p+1}
M is a finite abelian group isomorphic to (\mathbb{Z}/2\mathbb{Z})^{r(p + 1)} where r is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of M isomorphic to (\mathbb{Z}/2\mathbb{Z})^{r(p + 1)} where r is the rank (i.e., minimum number of generators) for the 2-Sylow subgroup of M
M is a finitely generated abelian group all isomorphic to (\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + s(p + 3)/2} where r is the rank for the 2-Sylow subgroup of the torsion part of M and s is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of M all isomorphic to (\mathbb{Z}/2\mathbb{Z})^{r(p + 1) + sp/2} where r is the rank for the 2-Sylow subgroup of M and s is the free rank (i.e., the rank as a free abelian group of the torsion-free part) of M

Cohomology groups and cohomology ring

Groups over the integers

The cohomology groups with coefficients in the integers are given as below:

H^p(\mathbb{Z}/2\mathbb{Z} \oplus \mathbb{Z}/2\mathbb{Z};\mathbb{Z}) = \left\lbrace \begin{array}{rl} (\mathbb{Z}/2\mathbb{Z})^{(p+1)/2}, & p = 1,3,5,\dots \\ (\mathbb{Z}/2\mathbb{Z})^{(p+2)/2}, & p = 2,4,6,\dots \\ \mathbb{Z}, & p = 0 \\\end{array}\right.

Second cohomology groups and extensions

Schur multiplier

The Schur multiplier, defined as the second cohomology group for trivial group action H^2(G,\mathbb{C}^\ast) and also as the second homology group H_2(G,\mathbb{Z}), is isomorphic to cyclic group:Z2.

See also the projective representation theory of Klein four-group.

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 four-group 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  ?  ?  ?