Second cohomology group for trivial group action of elementary abelian group of prime-square order on group of prime order

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Description of the group

Suppose p is a prime number.

We consider here the second cohomology group for trivial group action

\! H^2(G,A)

where G is the elementary abelian group of prime-square order E_{p^2} = \mathbb{Z}_p \times \mathbb{Z}_p and A is the group of prime order \mathbb{Z}_p.

The group is isomorphic to elementary abelian group of prime-cube order E_{p^3} = \mathbb{Z}_p \times \mathbb{Z}_p \times \mathbb{Z}_p. Equivalently, it is a three-dimensional vector space over the field \mathbb{F}_p of p elements.

The behavior is somewhat different for p = 2 and odd primes. For the anomalous case p = 2, see second cohomology group for trivial group action of V4 on Z2.

Computation in terms of group cohomology

The cohomology group can be computed as an abstract group using the group cohomology of elementary abelian group of prime-square order, which in turn can be computed using the Kunneth formula for group cohomology combined with the group cohomology of finite cyclic groups.

We explain here the part of the computation based on the group cohomology of elementary abelian group of prime-square order. As per that page, we have:

H^2(G;A) = (\operatorname{Ann}_A(p)) \oplus (A/pA)^2

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

In our case, A = \mathbb{Z}/p\mathbb{Z}, so we get that both A/pA and \operatorname{Ann}_A(p) are both isomorphic to \mathbb{Z}/p\mathbb{Z}. Plugging in, we get:

H^2(G;A) = \mathbb{Z}/p\mathbb{Z} \oplus (\mathbb{Z}/p\mathbb{Z})^2 = (\mathbb{Z}/p\mathbb{Z})^3

which is the elementary abelian group of order p^3.

Elements

Summary

FACTS TO CHECK AGAINST (second cohomology group for trivial group action):
Background reading on relationship with extension groups: Group extension problem
Arithmetic functions of extension group:
order (thus all extension groups have the same order): order of extension group is product of order of normal subgroup and quotient group
nilpotency class: nilpotency class of extension group is between nilpotency class of quotient group and one more for central extension
derived length: derived length of extension group is bounded by sum of derived length of normal subgroup and quotient group
minimum size of generating set: minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group|minimum size of generating set of quotient group is at most minimum size of generating set of group
WHAT'S THE TABLE BELOW?: Recall that there is a correspondence:
Elements of the group H^2(G;A) for the trivial group action \leftrightarrow congruence classes of central extensions with the specified subgroup A and quotient group G.
This descends to a correspondence:
Orbits for the group action of \operatorname{Aut}(G) \times \operatorname{Aut}(A) on H^2(G;A) \leftrightarrow pseudo-congruence classes of central extensions.
The table below breaks down the second cohomology group as a union of these orbits, with (as a general rule) each row describing one orbit, i.e., one "cohomology class type", aka one "pseudo-congruence class" of central extensions. The number of rows is the number of pseudo-congruence classes of central extensions.

As mentioned earlier, the information here does not apply to the case p = 2. For the case p = 2, see second cohomology group for trivial group action of V4 on Z2.

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is p^3) Stem extension? Base characteristic in whole group? Nilpotency class of whole group (at least 1, at most 2) Derived length of whole group (at least 1, at most 2) Minimum size of generating set of whole group (at least 2, at most 3) Subgroup information on base in whole group
trivial 1 elementary abelian group of prime-cube order 5 No No 1 1 3
symmetric and nontrivial p^2 - 1 direct product of cyclic group of prime-square order and cyclic group of prime order 2 No Yes 1 1 2
non-symmetric p - 1 unitriangular matrix group:UT(3,p) 3 Yes Yes 2 2 2 center of prime-cube order group:U(3,p)
non-symmetric (p - 1)(p^2 - 1) semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 Yes Yes 2 2 2 center of semidirect product of cyclic group of prime-square order and cyclic group of prime order
Total p^3 (equals order of the cohomology group) -- -- -- -- -- -- -- --

Explicit description and relation with power-commutator presentation

Further information: power-commutator presentation, presentations for groups of prime-cube order#Power-commutator presentations

Consider an extension group E with central subgroup isomorphic to A (group of prime order) and quotient group G isomorphic to elementary abelian group of prime-square order. Denote by \overline{a_1}, \overline{a_2} a basis for G (i.e., two non-identity elements of G that do not generate the same cyclic subgroup) and by a_1,a_2 elements of E that map to \overline{a_1},\overline{a_2} respectively. Denote by a_3 a non-identity element of the central subgroup.

Then, E is generated by the elements a_1,a_2,a_3. Further, we can get a power-commutator presentation for E using these generators. Specifically, we know that [a_1,a_3] = e, [a_2,a_3] = e, a_3^p = e. We also know that the elements a_1^p, a_2^p, [a_1,a_2] are each equal to some power of a_3.

In order to specify the cohomology class of the extension, it is necessary and sufficient to specify, for each of a_1^p, a_2^p, [a_1,a_2], what power of a_3 it equals. In terms of the notation for the power-commutator presentation, this is equivalent to saying that \beta(1,2) = 0 and each of \beta(1,3), \beta(2,3), \beta(1,2,3) can be represented as one of the numbers 0,1,2,\dots,p-1. This can be elements of \mathbb{Z}/p\mathbb{Z}, a concrete realization of the group of prime order. Here:

  • \beta(1,3) is the power of a_3 that a_1^p equals. It is 0 if a_1^p = e (i.e., is the identity element) and nonzero otherwise.
  • \beta(2,3) is the power of a_3 that a_2^p equals. It is 0 if a_2^p = e (i.e., is the identity element) and nonzero otherwise.
  • \beta(1,2,3) is the power of a_3 that [a_1,a_2] equals. It is 0 if [a_1,a_2] = e (i.e., is the identity element) and nonzero otherwise.

The total number of possibilities is p^3. Further, the mapping from H^2(G,A) that sends a cohomology class to the tuple (\beta(1,3),\beta(2,3),\beta(1,2,3)) is an isomorphism of additive groups. This means that to add two cohomology classes, we can add the corresponding tuples.

We provide below the full list of elements. Note that \beta(1,2) = 0 in all cases:

\beta(1,3) (equals 0 iff a_1^p = e) \beta(2,3) (equals 0 iff a_2^p = e) \beta(1,2,3) (equals 0 iff [a_1,a_2] = e) Number of such cohomology classes Isomorphism class of extension group Second part of GAP ID Explicit power-commutator presentation
zero zero zero 1 elementary abelian group of prime-cube order 5 [SHOW MORE]
nonzero zero zero p - 1 direct product of cyclic group of prime-square order and cyclic group of prime order 2 [SHOW MORE]
zero nonzero zero p - 1 direct product of cyclic group of prime-square order and cyclic group of prime order 2 [SHOW MORE]
nonzero nonzero zero (p - 1)^2 direct product of cyclic group of prime-square order and cyclic group of prime order 2 [SHOW MORE]
zero zero nonzero p - 1 unitriangular matrix group:UT(3,p) 3 [SHOW MORE]
nonzero zero nonzero (p - 1)^2 semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 [SHOW MORE]
zero nonzero nonzero (p - 1)^2 semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 [SHOW MORE]
nonzero nonzero nonzero (p - 1)^3 semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 [SHOW MORE]

Group actions

Action of automorphism group of acting group

By pre-composition, the automorphism group of the elementary abelian group of prime-square order (which is isomorphic to the general linear group of degree two over the field of p elements) acts on the second cohomology group. Under this action, there are a total of 2p orbits: each of the non-abelian extension types occurs in p - 1 orbits andthe abelian extension types occur in 1 orbit each. The details are given below:

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is p^3) Number of orbits under action of automorphism group of acting group Size of each orbit (must divide the order of the group, which is (p^2 - 1)(p^2 - p))
trivial 1 elementary abelian group of prime-cube order 5 1 1
symmetric and nontrivial p^2 - 1 direct product of cyclic group of prime-square order and cyclic group of prime order 2 1 p^2 - 1
non-symmetric p - 1 prime-cube order group:U(3,p) 3 p - 1 1
non-symmetric (p - 1)(p^2 - 1) semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 p - 1 p^2 - 1
Total p^3 (equals size of cohomology group) -- -- 2p --

Action of automorphism group of base group

By post-composition, the automorphism group of the group of prime order acts on the second cohomology group. Under this action, there are many orbits:

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is p^3) Number of orbits under action of automorphism group of base group Size of each orbit (must divide the order of the group, which is p - 1)
trivial 1 elementary abelian group of prime-cube order 5 1 1
symmetric and nontrivial p^2 - 1 direct product of cyclic group of prime-square order and cyclic group of prime order 2 p + 1 p - 1
non-symmetric p - 1 prime-cube order group:U(3,p) 3 1 p - 1
non-symmetric (p - 1)(p^2 - 1) semidirect product of cyclic group of prime-square order and cyclic group of prime order 4 p^2 - 1 p - 1
Total p^3 (size of cohomology group) -- -- p^2 + p + 2 (one more than size of projective two-dimensional space) --

Action of direct product of automorphism groups

If, however, we consider the action of the direct product of these groups, with one acting by pre-composition and the other by post-composition (since the actions are at opposite ends, they commute by associativity, so this is an action of the direct product), then the orbits are precisely the four cohomology class types described here, i.e., each cohomology class type gives one orbit.

Description of group action in terms of explicit descriptions of elements

The discussion here relies on the explicit description of cohomology classes in terms of the invariants \beta(1,3), \beta(2,3), \beta(1,2,3).

The automorphism group of the acting group, as noted above, is the general linear group GL(2,\mathbb{F}_p). It turns out that its action on \beta(1,3), \beta(2,3) is given precisely by the action as matrices, and it also induces an action on the \beta(1,2,3) by a group automorphism of \mathbb{Z}/p\mathbb{Z}.

The automorphism group of the base group, which is concretely \mathbb{F}_p^\ast, acts on all coordinates via multiplication by the corresponding element mod p. Thus, this is a scalar multiplication action. The orbits of non-identity elements are thus the lines in \mathbb{F}_p^3, or equivalently, the set of orbits can be described as two-dimensional projective space over \mathbb{F}_p. There is also the orbit of the identity or zero element, so the total number of orbits is (p^3-1)/(p-1) + 1 = p^2 + p + 1 + 1 = p^2 + p + 2.

Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action in terms of second homology group and abelianization#Case of abelian group

General background

We know from the general theory that there is a natural short exact sequence:

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G,A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(\bigwedge^2G,A) \to 0

where the image of \operatorname{Ext}^1 is H^2_{sym}(G;A), i.e., the group of cohomology classes represented by symmetric 2-cocycles and corresponding to the abelian group extensions. We also know, again from the general theory, that the short exact sequence above splits, i.e., H^2_{sym}(G;A) has a complement inside H^2. However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

Case of odd prime

Further information: Interpretation of Baer correspondence as natural splitting of short exact sequence from universal coefficients theorem

In this case, we can split the short exact sequence by considering the following one-sided inverse to the skew map: the map:

\! \operatorname{Hom}(\bigwedge^2G,A) \to H^2(G;A)

that sends an alternating bihomomorphism to the cohomology class represented by the 2-cocycle that equals the "half" of the bihomomorphism. Thus, we have a natural direct sum decomposition:

\! H^2(G;A) = H^2_{sym}(G,A) \oplus J

where J is the subgroup of H^2(G,A) comprising those cohomology classes that have as a representative an alternating bihomomorphism from G to A. In this case, H^2_{sym}(G,A) is an elementary abelian group of prime-square order and J is a group of prime order.

The \operatorname{Skew} map on H^2(G,A) that sends a cohomology class to the skew of any representative 2-cocycle, has H^2_{sym}(G,A) as its kernel and its effect on J is to output the alternating bihomomorphism that is twice of the alternating bihomomorphism representing the cohomology class.

This can also be interpreted in terms of the Baer correspondence. See the #Baer Lie rings section.

Case of the prime two

In this case, we do not get a direct sum decomposition of the above sort, because there is no notion of division by two or halving that would allow us to get an alternating bihomomorphism by halving the commutator map (which equals the skew of the 2-cocycle). For more, see second cohomology group for trivial group action of V4 on Z2#Direct sum decomposition.

Baer Lie rings

Odd prime case

The examples here illustrate the Baer correspondence

Recall from the discussion in the #Direct sum decomposition section that we have an internal direct sum decomposition:

\! H^2(G;A) = H^2_{\operatorname{sym}}(G;A) + J

where J is the subgroup of the second cohomology group comprising those classes that can be represented by alternating bihomomorphisms.

This can be used to obtain the Baer correspondence (the class two version of the Lazard correspondence) as follows: an element of H^2(G,A) corresponds to an extension group that is a group of nilpotency class two (which includes the abelian and non-abelian cases). It has a unique direct sum decomposition as the sum of a symmetric 2-cocycle (whose cohomology class is an element of H^2_{sym}(G,A)) and an alternating bihomomorphism (whose cohomology class is an element of J). The alternating bihomomorphism is given by halving the commutator map (which in turn is the skew of the 2-cocycle).

The Baer Lie ring for the group is defined as follows: the additive group is the extension corresponding to the symmetric 2-cocycle, and the Lie bracket is given by the alternating bihomomorphism.

Explicitly, for x,y in the extension group, we define the alternating bihomomorphism as \sqrt{[x,y]} and the addition as x + y := \frac{xy}{\sqrt{[x,y]}}.

We first make a picture of the cohomology group, where the top row is H^2_{\operatorname{sym}}, the left most column is J, and the remaining rows and columns are cosets of H^2_{\operatorname{sym}} and J respectively. The number of rows and columns depends on p, so when a row or column is repeated a certain number of times, we indicate the repetition in parentheses:

elementary abelian group of prime-cube order direct product of cyclic group of prime-square order and cyclic group of prime order (there are in fact p^2 - 1 such entries, each making its own column)
unitriangular matrix group:UT(3,p) (there are p - 1 such rows) semidirect product of cyclic group of prime-square order and cyclic group of prime order (this occurs in p^2 - 1 columns and p-1 rows).

The correspondence is between each group and the abelian group in its column (i.e., the top row group of its column). We thus have two nontrivial correspondences:

Group GAP ID Additive group of Lie ring GAP ID Information on Baer correspondence
unitriangular matrix group:UT(3,p) (p^3,3) elementary abelian group of prime-cube order (p^3,5) Baer correspondence between U(3,p) and u(3,p)
semidirect product of cyclic group of prime-square order and cyclic group of prime order (p^3,4) direct product of cyclic group of prime-square order and cyclic group of prime order (p^3,2) PLACEHOLDER FOR INFORMATION TO BE FILLED IN: [SHOW MORE]

Cocycles and coboundaries

Size information

We first give some quantitative size information if we use non-normalized cocycles and coboundaries:

Group Dimension as vector space over prime field Order of group (equals p to the power of dimension) Isomorphism class of group Explanation
group of 1-cocycles for trivial group action Z^1(G;A) 2 p^2 elementary abelian group of prime-square order Same as \operatorname{Hom}(G,A). see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
group of all 1-cochains for trivial group action C^1(G;A) 4 p^4 elementary abelian group of prime-fourth order all set maps from G to A with pointwise addition, so the dimension is the cardinality of G.
group of all 2-coboundaries for trivial group action B^2(G;A) 2 p^2 elementary abelian group of prime-square order By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all 2-cocycles for trivial group action Z^2(G;A) 5 p^5 elementary abelian group of prime-fifth order  ?
second cohomology group for trivial group action 3 p^3 elementary abelian group of prime-cube order This is Z^2/B^2, so dimensions subtract and orders divide.

In particular, what this means is that for every cohomology class, there are p^2 different choices of 2-cocycles that represent that cohomology class.

We give the corresponding information if we use normalized cocycles and coboundaries:

Group Dimension as vector space over field:F2 Order of group (equals 2 to the power of dimension) Isomorphism class of group Explanation
group of normalized 1-cocycles for trivial group action Z^1_n(G;A) 2 p^2 elementary abelian group of prime-square order Same as \operatorname{Hom}(G,A). see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
group of normalized 1-cochains for trivial group action C^1_n(G;A) 3 p^3 elementary abelian group of prime-cube order all set maps from G to A with pointwise addition, that send the identity to the identity. There are thus 3 elements that can be mapped arbitrarily.
group of all normalized 2-coboundaries for trivial group action B^2_n(G;A) 1 p group of prime order By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all normalized 2-cocycles for trivial group action Z^2_n(G;A) 4 p^4 elementary abelian group of prime-fourth order  ?
second cohomology group for trivial group action 3 p^3 elementary abelian group:E8 This is Z^2_n/B^2_n, so dimensions subtract and orders divide.

In particular, what this means is that for every cohomology class, there are p different choices of normalized 2-cocycles that represent that cohomology class.

Finding a group of cocycle representatives

Consider the short exact sequence for cocycles and coboundaries:

0 \to B^2(G;A) \to Z^2(G;A) \to H^2(G;A) \to 0

and the corresponding one for normalized cocycles and coboundaries:

0 \to B^2_n(G;A) \to Z^2_n(G;A) \to H^2(G;A) \to 0

Since these are short exact sequences of vector spaces, they must split. Further, a splitting of the latter also gives a splitting of the former.

GAP implementation

Construction of the cohomology group

The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF. Begin by setting p equal to a specific prime number value.

gap> G := ElementaryAbelianGroup(p^2);;
gap> A := TrivialGModule(G,GF(p));;
gap> T := TwoCohomology(G,A);

The precise output depends on the value of p.

Construction of extensions

The extensions can be constructed using the additional command Extensions. Begin by setting p equal to a specific prime number value.

gap> G := ElementaryAbelianGroup(p^2);;
gap> A := TrivialGModule(G,GF(p));;
gap> L := Extensions(G,A);;
gap> List(L,IdGroup);

The precise output depends on the value of p.

Construction of automorphism group actions

This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives. Begin by setting p equal to a specific prime number value.

gap> G := ElementaryAbelianGroup(p^2);;
gap> A := TrivialGModule(G,GF(p));;
gap> A1 := AutomorphismGroup(G);;
gap> A2 := GL(1,p);;
gap> D := DirectProduct(A1,A2);;
gap> P := CompatiblePairs(G,A,D);;
gap> M := ExtensionRepresentatives(G,A,P);;
gap> List(M,IdGroup);