Difference between revisions of "Group cohomology of elementary abelian group of prime-square order"

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(Over an abelian group)
(Over an abelian group)
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===Over an abelian group===
 
===Over an abelian group===
  
The homology groups with coefficients in an abelian group <math>M</math> are given as follows:
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The homology groups with coefficients in an abelian group <math>M</math> are given as follows. Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>:
  
 
<math>\! H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/pM)^{(q+3)/2} \oplus (\operatorname{Ann}_M(p))^{(q-1)/2}, & \qquad q = 1,3,5,\dots\\ (M/pM)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0\\\end{array}\right.</math>
 
<math>\! H_q(\mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z};M) = \left\lbrace\begin{array}{rl} (M/pM)^{(q+3)/2} \oplus (\operatorname{Ann}_M(p))^{(q-1)/2}, & \qquad q = 1,3,5,\dots\\ (M/pM)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad q = 0\\\end{array}\right.</math>
  
Here, <math>M/pM</math> is the quotient of <math>M</math> by <math>pM = \{ px \mid x \in M \}</math> and <math>\operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}</math>.
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These homology groups can be computed in terms of the homology groups over integers using the [[universal coefficients theorem for group homology]].
 
These homology groups can be computed in terms of the homology groups over integers using the [[universal coefficients theorem for group homology]].

Revision as of 16:39, 24 October 2011

Suppose p is a prime number. We are interested in the elementary abelian group of prime-square order E_{p^2} = (\mathbb{Z}/p\mathbb{Z})^2 = \mathbb{Z}/p\mathbb{Z} \oplus \mathbb{Z}/p\mathbb{Z}.

Particular cases

Value of prime p elementary abelian group of prime-square order cohomology information
2 Klein four-group group cohomology of Klein four-group
3 elementary abelian group:E9 group cohomology of elementary abelian group:E9
5 elementary abelian group:E25 group cohomology of elementary abelian group:E25

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

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

The first few homology groups are given below:

q 0 1 2 3 4 5
H_q \mathbb{Z} (\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2} \mathbb{Z}/p\mathbb{Z} (\mathbb{Z}/p\mathbb{Z})^3 = E_{p^3} (\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2} (\mathbb{Z}/p\mathbb{Z})^4 = E_{p^4}
rank of H_q as an elementary abelian p-group -- 2 1 3 2 4

Over an abelian group

The homology groups with coefficients in an abelian group M are given as follows. Here, M/pM is the quotient of M by pM = \{ px \mid x \in M \} and \operatorname{Ann}_M(p) = \{ x \in M \mid px = 0 \}:

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


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 M Conclusion about odd-indexed homology groups, i.e., H_q, q = 1,3,5,\dots Conclusion about even-indexed homology groups, i.e., H_q, q = 2,4,6,\dots
M is uniquely p-divisible, i.e., every element of M can be divided uniquely by p. This includes the case that M is a field of characteristic not p. all zero groups all zero groups
M is p-torsion-free, i.e., no nonzero element of M multiplies by p to give zero. (M/pM)^{(q+3)/2} (M/pM)^{q/2}
M is p-divisible, but not necessarily uniquely so, e.g., M = \mathbb{Q}/\mathbb{Z} (\operatorname{Ann}_M(p))^{(q-1)/2} (\operatorname{Ann}_M(p))^{(q+2)/2}
M = \mathbb{Z}/p^n\mathbb{Z}, n any natural number (\mathbb{Z}/p\mathbb{Z})^{q+1} (\mathbb{Z}/p\mathbb{Z})^{q+1}
M is a finite abelian group isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(q + 1)} where r is the rank (i.e., minimum number of generators) for the p-Sylow subgroup of M isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(q + 1)} where r is the rank (i.e., minimum number of generators) for the p-Sylow subgroup of M
M is a finitely generated abelian group all isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2} where r is the rank for the p-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}/p\mathbb{Z})^{r(q + 1) + sq/2} where r is the rank for the p-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 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

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

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

The first few cohomology groups are given below:

q \! 0 \! 1 \! 2 \! 3 \! 4 \! 5
H^q \mathbb{Z} 0 (\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2} \mathbb{Z}/p\mathbb{Z} (\mathbb{Z}/p\mathbb{Z})^3 = E_{p^3} (\mathbb{Z}/p\mathbb{Z})^2 = E_{p^2}
rank of H^q as an elementary abelian p-group -- 0 2 1 3 2

Over an abelian group

The cohomology groups with coefficients in an abelian group M are given as follows:

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

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

These can be deduced from the homology groups with coefficients in the integers using the dual universal coefficients theorem for group cohomology.

Important case types for abelian groups

Important case types for abelian groups

Case on M Conclusion about odd-indexed cohomology groups, i.e., H^q, q = 1,3,5,\dots Conclusion about even-indexed homology groups, i.e., H^q, q = 2,4,6,\dots
M is uniquely p-divisible, i.e., every element of M can be divided by p uniquely. This includes the case that M is a field of characteristic not 2. all zero groups all zero groups
M is p-torsion-free, i.e., no nonzero element of M multiplies by p to give zero. (M/pM)^{(q-3)/2} (M/pM)^{(q+2)/2}
M is p-divisible, but not necessarily uniquely so, e.g., M = \mathbb{Q}/\mathbb{Z} (\operatorname{Ann}_M(p))^{(q+3)/2} (\operatorname{Ann}_M(p))^{q/2}
M = \mathbb{Z}/p^n\mathbb{Z}, n any natural number (\mathbb{Z}/p\mathbb{Z})^{q+1} (\mathbb{Z}/p\mathbb{Z})^{q+1}
M is a finite abelian group isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(q + 1)} where r is the rank (i.e., minimum number of generators) for the p-Sylow subgroup of M isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(p + 1)} where r is the rank (i.e., minimum number of generators) for the p-Sylow subgroup of M
M is a finitely generated abelian group all isomorphic to (\mathbb{Z}/p\mathbb{Z})^{r(q + 1) + s(q - 1)/2} where r is the rank for the p-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}/p\mathbb{Z})^{r(q + 1) + s(q + 3)/2} where r is the rank for the p-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