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

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(Over an abelian group)
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The homology groups with coefficients in an abelian group <math>M</math> are given as follows:
 
The homology groups with coefficients in an abelian group <math>M</math> are given as follows:
  
<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/2M)^{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>.
 
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>.

Revision as of 19:04, 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

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:

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/2M)^{q/2} \oplus (\operatorname{Ann}_M(p))^{(q+2)/2}, & \qquad q = 2,4,6,\dots \\ M, & \qquad 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 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

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

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