Second cohomology group for trivial group action of V4 on Z2

This article gives information about the second cohomology group for trivial group action (i.e., the second cohomology group with trivial action) of the group Klein four-group on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and Klein four-group the corresponding quotient group. Specifically, these are precisely the central extensions with the given base group and acting group.
The value of this cohomology group is elementary abelian group:E8.
Get more specific information about Klein four-group |Get more specific information about cyclic group:Z2|View other constructions whose value is elementary abelian group:E8

Description of the group

We consider here the second cohomology group for trivial group action of the Klein four-group on cyclic group:Z2, i.e.,



where  and .

The cohomology group is isomorphic to elementary abelian group:E8.

Computation in terms of group cohomology

The cohomology group can be computed as an abstract group using the group cohomology of Klein four-group, which in turn can be computed using the Kunneth formula for group cohomology combined with the group cohomology of cyclic group:Z2.

We explain here the part of the computation based on the group cohomology of Klein four-group. As per that page, we have:



Here,  is the quotient of  by  and .

In our case, , so we get that both  and  are also . Plugging in, we get:



which is the elementary abelian group of order eight.

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  for the trivial group action  congruence classes of central extensions with the specified subgroup  and quotient group .
This descends to a correspondence:
Orbits for the group action of  on   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.

Each element of the second cohomology group corresponds to a group extension with base normal subgroup cyclic group:Z2 in the center and the quotient group isomorphic to Klein four-group. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each extension group is .

Further, the minimum size of generating set of the extension group is at least equal to 2 (the minimum size of generating set of the quotient Klein four-group) and at most equal to 3 (the sum of the minimum size of generating set for the normal subgroup and quotient group).

Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 8) Stem extension? Base characteristic in whole group? Hall-Senior family (equivalence class up to being isoclinic) Nilpotency class of whole group Derived length of whole group Minimum size of generating set of whole group (must be at least 2, at most 3) Subgroup information on base in whole group
trivial 1 elementary abelian group:E8 5 No No  1 1 3 Z2 in E8
symmetric and nontrivial 3 direct product of Z4 and Z2 2 No Yes  1 1 2 first agemo subgroup of direct product of Z4 and Z2
non-symmetric 3 dihedral group:D8 3 Yes Yes  2 2 2 center of dihedral group:D8
non-symmetric 1 quaternion group 4 Yes Yes  2 2 2 center of quaternion group
Total (4 rows) 8 (= order of the cohomology group) -- -- -- -- -- -- -- -- --

Explicit description and relation with power-commutator presentation

Further information: presentations for groups of order 8#Power-commutator presentations

Consider an extension group  with central subgroup isomorphic to  (cyclic group:Z2) and quotient group  isomorphic to Klein four-group. Denote by  a basis for  (i.e., two unequal non-identity elements of ) and by  elements of  that map to  respectively. Denote by  a non-identity element of the central subgroup.

Then,  is generated by the elements . Further, we can get a power-commutator presentation for  using these generators. Specifically, we know that . We also know that the elements  are each either equal to  or to .

In order to specify the cohomology class of the extension, it is necessary and sufficient to specify, for each of , whether it equals  or . In terms of the notation for the power-commutator presentation, this is equivalent to saying that  and each of  can be either 0 or 1, and are viewed as elements of cyclic group:Z2. Here:

•  is the power of  that  equals. It is 0 if  (i.e., is the identity element) and 1 if .
•  is the power of  that  equals. It is 0 if  (i.e., is the identity element) and 1 if .
•  is the power of  that  equals. It is 0 if  (i.e., is the identity element) and 1 if .

The total number of possibilities is . Further, the mapping from  that sends a cohomology class to the tuple  is an isomorphism of additive groups. This means that to add two cohomology classes, we can add the corresponding tuples.

Here is a more detailed explanation for why the mapping described here is well defined up to cohomology and gives an isomorphism:

We provide below the full list of elements. Note that  in all cases:

 (equals 0 iff  equals 1 iff )  (equals 0 iff , equals 1 iff )  (equals 0 iff , equals 1 iff ) Isomorphism class of extension group Second part of GAP ID Explicit power-commutator presentation
1 0 0 direct product of Z4 and Z2 2 [SHOW MORE]
0 1 0 direct product of Z4 and Z2 2 [SHOW MORE]
1 1 0 direct product of Z4 and Z2 2 [SHOW MORE]

Group actions

Summary of action

By pre-composition, the automorphism group of the Klein four-group (which is isomorphic to symmetric group:S3) acts on the second cohomology group. Under this action, there are four orbits, corresponding to the four types of group extensions given above. The elementary abelian group:E8 and quaternion group are fixed points. Specifically, the 3 cohomology classes that give direct product of Z4 and Z2 are in one orbit, while the 3 cohomology classes that give dihedral group:D8 are in another 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 .

The automorphism group of the Klein four-group acts as symmetric group:S3 on the elements . This shuffling leads to a corresponding change in the values of  and . The invariant , which determines whether the extension group is abelian, is invariant under this transformation. However, the value of  affects the way in which the automorphism affects  and .

Subgroups of interest

Subgroup Quotient group (i.e., what each coset signifies) Value as group Condition in terms of explicit description by power-commutator presentation Corresponding group extensions GAP IDs second part Group extension groupings for each coset GAP IDs second part
subgroup generated by images of symmetric 2-cocycles (corresponds to abelian group extensions) second cohomology group up to isoclinism Klein four-group  elementary abelian group:E8 and direct product of Z4 and Z2 (3 times) 5,2 (elementary abelian group:E8, direct product of Z4 and Z2) (1 copy) and (quaternion group and dihedral group:D8) (1 copy) (5,2) (1 copy) and (4,3) (1 copy)
IIP subgroup of second cohomology group for trivial group action trivial group  elementary abelian group:E8 only 5 no grouping occurs, all cosets have size one 2,3,4,5
cyclicity-preserving subgroup of second cohomology group for trivial group action trivial group  elementary abelian group:E8 only 5 no grouping occurs, all cosets have size one 2,3,4,5

Application to other extensions

Trivial outer action

We consider the classification of congruence classes of extensions where the base normal subgroup has center isomorphic to cyclic group:Z2 and the quotient group is isomorphic to the Klein four-group, and the quotient has trivial outer action on the base. In this case, the set of congruence classes of extensions is classified by , with the identity element corresponding to the external direct product.

Group with center  Order GAP ID Information on extensions Order of extensions trivial group extension GAP ID (2nd part) symmetric and nontrivial group extension GAP ID non-symmetric extension that gives  in the base case GAP ID (2nd part) non-symmetric extension that gives  in the base case GAP ID
dihedral group:D8 8 3 extensions for trivial outer action of V4 on D8 32 direct product of D8 and V4 46 direct product of SmallGroup(16,13) and Z2 48 inner holomorph of D8 49 central product of D8 and Q8 50
quaternion group 8 4 extensions for trivial outer action of V4 on Q8 32 direct product of Q8 and V4 47 direct product of SmallGroup(16,13) and Z2 48 central product of D8 and Q8 50 inner holomorph of D8 49

Direct sum decomposition

For background information, see formula for second cohomology group for trivial group action of abelian group in terms of Schur multiplier and abelianization

General background

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



where the image of  is , i.e., the group of cohomology classes represented by symmetric 2-cocycles. We also know, again from the general theory, that the short exact sequence above splits, i.e.,  has a complement inside . However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case

For this choice of  and , the subgroup  of cyclicity-preserving cohomology classes is trivial, hence the sum  is  and is not the whole group . Thus,  is not the desired complement.

However, there is a complement that we call  to  in . This comprises the trivial extension and the extension that gives the quaternion group. Further, this complement is invariant under the action of . As an internal direct sum:



as an internal direct sum. A pictorial description of this would be as follows. Here, each column is a coset of  and each row is a coset of . The top left entry is the identity element, hence the top row corresponds to abelian group extensions and the left column corresponds to .

The group  acts as symmetric group:S3 on the three columns other than the left column. Each of the rows is preserved.

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 field:F2 Order of group (equals 2 to the power of dimension) Isomorphism class of group Explanation
group of 1-cocycles for trivial group action  2 4 Klein four-group Same as . see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
group of all 1-cochains for trivial group action  4 16 elementary abelian group:E16 all set maps from  to  with pointwise addition, so the dimension is the cardinality of .
group of all 2-coboundaries for trivial group action  2 4 Klein four-group 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  5 32 elementary abelian group:E32  ?
second cohomology group for trivial group action 3 8 elementary abelian group:E8 This is , so dimensions subtract and orders divide.

In particular, what this means is that for every cohomology class, there are 4 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  2 4 Klein four-group Same as . see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
group of normalized 1-cochains for trivial group action  3 8 elementary abelian group:E8 all set maps from  to  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  1 2 cyclic group:Z2 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  4 16 elementary abelian group:E16  ?
second cohomology group for trivial group action 3 8 elementary abelian group:E8 This is , so dimensions subtract and orders divide.

In particular, what this means is that for every cohomology class, there are 2 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:



and the corresponding one for normalized cocycles and coboundaries:



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.

There is no automorphism-invariant choice of splitting (i.e., no choice of splitting that is invariant under the -action). However, we can define splittings once we have established a basis for .

Use the same notation as in the section on explicit descriptions and power-commutator presentations. Thus,  has basis  and  has elements  with  the identity element and  the non-identity element. Then, for each cohomology class, pick the unique representative normalized 2-cocycle  such that:



or, if we denote the group additively with :



What this basically says, in terms of coset representative choices, is that we are making sure that the coset representative for  equals . In other words, we are pinning down the ambiguity in picking one of the non-identity coset representatives. Our choice of normalized already eliminates the ambiguity with picking the identity coset representative. The computations above show that this picks a unique coset representative.

Further, by the pointwise nature of cocycle addition, these uniquely chosen coset representatives form a subgroup.

Linear choices of cocycle and algebra group structures

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  2 4 Klein four-group Same as . see first cohomology group for trivial group action is naturally isomorphic to group of homomorphisms
subgroup of group of normalized 1-cochains for trivial group action  whose coboundary is linear. 3 8 elementary abelian group:E8 all set maps from  to  with pointwise addition, that send the identity to the identity. There are thus 3 elements that can be mapped arbitrarily.
group of all bilinear 2-coboundaries  1 2 cyclic group:Z2 By the first isomorphism theorem and the definition of this group, it is isomorphic to the group (1-cochains with linear coboundary)/(1-cocycles), so the dimensions as vector spaces subtract and the orders divide.
group of all bilinear maps , a subgroup of the group of normalized 2-cocycles  4 16 elementary abelian group:E16 Since  is one-dimensional, the dimension is the square of the dimension of , which is .
subgroup second cohomology group for trivial group action that has a cocycle representative that is linear 3 8 elementary abelian group:E8 This is the quotient of the group of all bilinear maps by the subgroup of bilinear 2-coboundaries, so the orders divide.

The upshot is that all the cohomology classes can be represented by bilinear choices of 2-cocycle, and there are two choices for each cohomology class. A bilinear choice of 2-cocycle means that the group has the structure of an adjoint group for the corresponding radical ring. In this case, because everything is happening over field:F2, we get algebra group structures over the field. To summarize: there are two algebra group structures for the extension group corresponding to each cohomology class. In all the algebra group structures, the corresponding algebra has nilpotency index at most three, i.e., all products of length three in each of the algebras is zero.

Note that the extension groups may have other algebra groups unrelated to their realization in terms of these cohomology classes.

Homomorphisms to and from other cohomology groups

Homomorphisms on 

The unique injective homomorphism from  to  induces a homomorphism:



The group on the right is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of V4 on Z4). However, the induced map above is not an isomorphism. Rather, it has kernel  (comprising all the abelian extensions) and its image is the cyclicity-preserving subgroup of second cohomology group for trivial group action  that comprises the trivial group extension (direct product of Z4 and V4) and the extension central product of Q8 and Z4.

In terms of extensions, the map is interpreted as follows: it involves taking the central product of a given extension with cyclic group:Z4, identifying the base cyclic group:Z2 in the original extension with the  in .

The map is given in the table below.

Input Number of copies Output = central product of input group with  over identified central subgroup 
elementary abelian group:E8 1 direct product of Z4 and V4
direct product of Z4 and Z2 3 direct product of Z4 and V4
dihedral group:D8 3 central product of D8 and Z4
quaternion group 1 central product of D8 and Z4

The unique surjective homomorphism from  to  induces a homomorphism:



The kernel of this map is the cyclicity-preserving subgroup of second cohomology group for trivial group action  that comprises the trivial group extension (direct product of Z4 and V4) and the extension central product of Q8 and Z4. The image is the group . Thus, the roles of kernel and image are reversed from the previous map.

Input Number of copies Output
direct product of Z4 and V4 1 elementary abelian group:E8
direct product of Z8 and Z2 3 direct product of Z4 and Z2
M16 3 direct product of Z4 and Z2
central product of D8 and Z4 1 elementary abelian group:E8

Homomorphism on 

For each injective map from  to , we get a corresponding restriction homomorphism:



The kernel of each of these homomorphisms is a Klein four-group comprising one copy of elementary abelian group:E8, one copy of direct product of Z4 and Z2, and two copies of dihedral group:D8. Each of the Klein four-groups is different and  permutes them transitively. The only element not in any of the kernels is the element corresponding to the quaternion group extension.

For each surjective map from  to , we get a corresponding inflation homomorphism:



For each of these, the image of the homomorphism is a cyclic group:Z2 comprising the trivial extension elementary abelian group:E8 and one copy of the symmetric nontrivial extension direct product of Z4 and Z2. Each of these image subgroups is different, and their union is the whole of . The automorphism group  permutes them transitively.

GAP implementation

Construction of the cohomology group

The cohomology group can be constucted using the GAP functions ElementaryAbelianGroup, TwoCohomology, TrivialGModule, GF.

```gap> G := ElementaryAbelianGroup(4);;
gap> A := TrivialGModule(G,GF(2));;
gap> T := TwoCohomology(G,A);
rec( group := <pc group of size 4 with 2 generators>,
module := rec( field := GF(2), isMTXModule := true, dimension := 1,
generators := [ <an immutable 1x1 matrix over GF2>,
<an immutable 1x1 matrix over GF2> ] ),
collector := rec( relators := [ [ 0 ], [ [ 2, 1 ], 0 ] ],
orders := [ 2, 2 ], wstack := [ [ 1, 1 ], [ 2, 1 ] ], estack := [  ],
pstack := [ 3, 3 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ],
list := [ 0, 0 ],
module := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x
1 matrix over GF2> ], mone := <an immutable 1x1 matrix over GF2>,
mzero := <an immutable 1x1 matrix over GF2>, avoid := [  ],
unavoidable := [ 1, 2, 3 ] ),
cohom := <linear mapping by matrix, <vector space of dimension 3 over GF(
2)> -> ( GF(2)^3 )>,
presentation := rec( group := <free group on the generators [ f1, f2 ]>,
relators := [ f1^2, f1^-1*f2*f1*f2^-1, f2^2 ] ) )```

Construction of extensions

The extensions can be constructed using the additional command Extensions.

```gap> G := ElementaryAbelianGroup(4);;
gap> A := TrivialGModule(G,GF(2));;
gap> L := Extensions(G,A);;
gap> List(L,IdGroup);
[ [ 8, 5 ], [ 8, 2 ], [ 8, 3 ], [ 8, 3 ], [ 8, 2 ], [ 8, 2 ], [ 8, 3 ],
[ 8, 4 ] ]
gap> FrequencySort(last);
[ [ [ 8, 2 ], 3 ], [ [ 8, 3 ], 3 ], [ [ 8, 4 ], 1 ], [ [ 8, 5 ], 1 ] ]```

Construction of automorphism group actions

This uses additionally the GAP functions AutomorphismGroup, DirectProduct, CompatiblePairs, and ExtensionRepresentatives.

```gap> G := ElementaryAbelianGroup(4);;
gap> A := TrivialGModule(G,GF(2));;
gap> A1 := AutomorphismGroup(G);;
gap> A2 := GL(1,2);;
gap> D := DirectProduct(A1,A2);;
gap> P := CompatiblePairs(G,A,D);;
gap> M := ExtensionRepresentatives(G,A,P);;
gap> List(M,IdGroup);
[ [ 8, 5 ], [ 8, 2 ], [ 8, 3 ], [ 8, 4 ] ]```