Second cohomology group for trivial group action of D8 on Z2

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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 dihedral group:D8 on cyclic group:Z2. The elements of this classify the group extensions with cyclic group:Z2 in the center and dihedral group:D8 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 dihedral group:D8 |Get more specific information about cyclic group:Z2|View other constructions whose value is elementary abelian group:E8

Definition

This article is about the second cohomology group for trivial group action where the acting group is dihedral group:D8 (the dihedral group of order 8 and degree 4) and the base group is cyclic group:Z2 (the cyclic group of order 2). In other words, we are interested in:

H^2(G,A)

where G \cong D_8 and A \cong \mathbb{Z}_2.

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

Computation of cohomology group

The cohomology group can be computed as an abstract group using group cohomology of dihedral group:D8.

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.


Cohomology class type Number of cohomology classes Corresponding group extension GAP ID (second part, order is 16) Stem extension? Base characteristic in whole group? Hall-Senior family (equivalence class up to being isoclinic) Nilpotency class of whole group (at least 2, at most 3) Derived length of whole group (always exactly 2) Minimum size of generating set of whole group (at least 2, at most 3) Subgroup information on base in whole group
trivial 1 direct product of D8 and Z2 11 No No \Gamma_2 2 2 3
nontrivial and symmetric 1 nontrivial semidirect product of Z4 and Z4 4 No Yes \Gamma_2 2 2 2 Subgroup generated by a non-commutator square in nontrivial semidirect product of Z4 and Z4
nontrivial and symmetric 2 SmallGroup(16,3) 3 No No \Gamma_2 2 2 2
non-symmetric 1 dihedral group:D16 7 Yes Yes \Gamma_3 3 2 2 center of dihedral group:D16
non-symmetric 1 generalized quaternion group:Q16 9 Yes Yes \Gamma_3 3 2 2 center of generalized quaternion group:Q16
non-symmetric 2 semidihedral group:SD16 8 Yes Yes \Gamma_3 3 2 2 center of semidihedral group:SD16
Total (6 rows) 8 (equals order of the cohomology group) -- -- -- -- -- -- -- -- --


Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z2 and the quotient group isomorphic to dihedral group:D8. Due to the fact that order of extension group is product of order of normal subgroup and quotient group, the order of each of the extension groups is 2 \times 8 = 16.

Some, but not all, of the extensions are stem extensions. Since cyclic group:Z2 is in fact also the Schur multiplier of dihedral group:D8, the stem extensions here are precisely the Schur covering groups of dihedral group:D8.

The minimum size of generating set of the extension group is at least equal to 2 (which is the minimum size of generating set of the quotient group) and at most equal to 3 (which is the sum of the minimum size of generating set of the normal subgroup and the quotient group). See minimum size of generating set of extension group is bounded by sum of minimum size of generating set of normal subgroup and quotient group and minimum size of generating set of quotient group is at most minimum size of generating set of group.

The nilpotency class is at least 2 and at most 3 in all cases. It is at least 2 because the quotient dihedral group:D8 has nilpotency class two. It is at most 3 because the sum of the nilpotency class of the normal subgroup and quotient group is 3, and the extension is a central extension. The derived length is always exactly 2 because nilpotency class 2 or 3 forces derived length exactly 2, using derived length is logarithmically bounded by nilpotency class.

Explicit description and relation with power-commutator presentation

Further information: power-commutator presentation, presentations for groups of order 16#Power-commutator presentations

Consider an extension group E with central subgroup isomorphic to A (cyclic group:Z2) and quotient group G isomorphic to dihedral group:D8. Denote by \overline{a_1}, \overline{a_2}, \overline{a_3} elements of G such that \overline{a_1}^2 = \overline{a_3}, \overline{a_2} has order two, and [\overline{a_1},\overline{a_2}] = \overline{a_3}. Explicitly, we are considering G as given by the presentation \langle \overline{a_1}, \overline{a_2}, \overline{a_3} \mid \overline{a_1}^2 = \overline{a_3}, \overline{a_2}^2 = e, \overline{a_3}^2 = e, [\overline{a_1},\overline{a_2}] = \overline{a_3}, [\overline{a_1},\overline{a_3}] = e, [\overline{a_2},\overline{a_3}] = e \rangle.

We pick elements a_1,a_2,a_3 of E that live in the cosets corresponding to \overline{a_1}, \overline{a_2}, \overline{a_3}, with the additional condition that a_3 = a_1^2. Next, we pick a_4 as the non-identity element of A.

We can now write a power-commutator presentation of E in terms of a_1,a_2,a_3,a_4. With the usual notation, we already have \beta(1,3) = \beta(1,2,3) = 1, and all the other coefficients involving 1-3 are zero. This leaves us to choose the values \beta(1,4), \beta(2,4), \beta(3,4), \beta(1,2,4), \beta(1,3,4), \beta(2,3,4). However, we note that \beta(1,4) = \beta(1,3,4) = 0 because in our specific choice, a_3 equals a_1^2. Thus, we in fact have only four values to choose: \beta(2,4), \beta(3,4), \beta(1,2,4), \beta(2,3,4).

Further, it turns out that of these four values, we must have \beta(3,4) = \beta(2,3,4) under these conditions. To see this, note that if [a_3,a_2] = [a_1^2,a_2]. Because a_1 commutes with [a_1,a_2] this can be rewritten as [a_1,a_2]^2. Note that [a_1,a_2] = a_3 or [a_1,a_2] = a_3a_4, so its square is a_3^2. The upshot is that [a_3,a_2] = a_3^2. Inverting, we get [a_2,a_3] = a_3^2.

The upshot of all this is that we can freely vary the parameters \beta(2,4), \beta(3,4), \beta(1,2,4) and the remaining parameters are fixed. This gives 2^3 = 8 possibilities. Moreover, the coordinate-wise addition of these corresponds to addition in the cohomology group.

Note that the positions of dihedral group:D16 and semidihedral group:SD16 are sensitive to the commutator convention.

\beta(2,4) \beta(1,2,4) \beta(3,4) = \beta(2,3,4) Corresponding extension group Second part of GAP ID (order is 16) Nilpotency class
0 0 0 direct product of D8 and Z2 11 2
1 0 0 nontrivial semidirect product of Z4 and Z4 4 2
0 1 0 SmallGroup(16,3) 3 2
1 1 0 SmallGroup(16,3) 3 2
0 0 1 dihedral group:D16 7 3
1 0 1 generalized quaternion group:Q16 9 3
0 1 1 semidihedral group:SD16 8 3
1 1 1 semidihedral group:SD16 8 3

Group actions

Under the action of the automorphism group of the acting group

By pre-composition, the automorphism group of dihedral group:D8, which is itself isomorphic to dihedral group:D8, acts on the second cohomology group. This action is transitive on all the extensions in each cohomology class type. In particular, the four fixed points are the cohomology classes corresponding to the extension groups direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, dihedral group:D16 and generalized quaternion group:Q16, and the two orbits of size two correspond to the extensions SmallGroup(16,3) and semidihedral group:SD16.

Subgroups of interest

Subgroup Quotient Value Corresponding group extensions GAP IDs second part Group extension groupings for each coset GAP IDs second part
second cohomology group restricted to subvariety of abelian groups: subgroup generated by images of symmetric 2-cocycles, equivalently, image of \operatorname{Ext}^1(G^{\operatorname{ab}},A) in the direct sum decomposition arising from the universal coefficients theorem (this means that the restriction to the derived subgroup of the acting group gives a split extension and all the interesting stuff is happening at the level of the abelianization of the acting group) second cohomology group up to isoclinism Klein four-group direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, SmallGroup(16,3) (2 times) 11,4,3 (direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, SmallGroup(16,3) (2 times)), (dihedral group:D16, generalized quaternion group:Q16, semidihedral group:SD16 (2 times)) (11,4,3), (7,9,8)
second cohomology group restricted to class two groups second cohomology group up to isologism for groups of nilpotency class two Klein four-group direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, SmallGroup(16,3) (2 times) 11,4,3 (direct product of D8 and Z2, nontrivial semidirect product of Z4 and Z4, SmallGroup(16,3) (2 times)), (dihedral group:D16, generalized quaternion group:Q16, semidihedral group:SD16 (2 times)) (11,4,3), (7,9,8)

Direct sum decomposition

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

General background

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

0 \to \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) \to H^2(G;A) \stackrel{\operatorname{Skew}}{\to} \operatorname{Hom}(H_2(G;\mathbb{Z}),A) \to 0

Here:

  • The group \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) corresponds to the subgroup H^2_{\operatorname{sym}}(G;A) of cohomology classes represented by symmetric 2-cocycles. G^{\operatorname{ab}} is the abelianization of G and its image comprises those extensions where the restricted extension of the derived subgroup [G,G] on A is trivial and the corresponding extension of the quotient group is abelian.
  • The group \operatorname{Hom}(H_2(G;\mathbb{Z}),A) is the second cohomology group up to isoclinism: Explicitly, H_2(G;\mathbb{Z}) is the Schur multiplier of G.

We also know, again from the general theory, that the short exact sequence above splits, i.e., the image of \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) in H^2(G;A) has a complement inside H^2(G;A). However, there need not in general be a natural or even an automorphism-invariant choice of splitting.

In this case

For this choice of G and A, G^{\operatorname{ab}} is isomorphic to the Klein four-group. The corresponding group \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) is also a Klein four-group.

The Schur multiplier H_2(G;\mathbb{Z}) is cyclic group:Z2, hence \operatorname{Hom}(H_2(G;\mathbb{Z}),A) is also isomorphic to cyclic group:Z2.

The image of \operatorname{Ext}^1_{\mathbb{Z}}(G^{\operatorname{ab}},A) in H^2(G;A) comprises the four non-stem extensions. It has two cosets in the whole second cohomology group. To split the short exact sequence in an automorphism-invariant fashion, we could pick as our complement either dihedral group:D16 or generalized quaternion group:Q16. The two possibilities are shown below:

Using dihedral group:D16 as the choice of complement, where the rows represent the cosets of the image of \operatorname{Ext}^1, and the columns represent the cosets of the complement:

direct product of D8 and Z2 nontrivial semidirect product of Z4 and Z4 SmallGroup(16,3) SmallGroup(16,3)
dihedral group:D16 generalized quaternion group:Q16 semidihedral group:SD16 semidihedral group:SD16

Using generalized quaternion group:Q16 as the choice of complement, where the rows represent the cosets of the image of \operatorname{Ext}^1, and the columns represent the cosets of the complement:

direct product of D8 and Z2 nontrivial semidirect product of Z4 and Z4 SmallGroup(16,3) SmallGroup(16,3)
generalized quaternion group:Q16 dihedral group:D16 semidihedral group:SD16 semidihedral group:SD16

More on the mapping from \operatorname{Ext}^1

Below is an explicit description of the mapping from \operatorname{Ext}^1(G^{\operatorname{ab}},A) to H^2(G;A), in terms of the original extension for Klein four-group (the abelianization of dihedral group:D8) on top of cyclic group:Z2 and what the new extension looks like for dihedral group:D8 on top of cyclic group:Z2.

Cohomology class type Number of such cohomology classes Element of \operatorname{Ext}^1 as an extension Image in H^2(G;A)
trivial 1 elementary abelian group:E8 direct product of D8 and Z2
nontrivial 1 direct product of Z4 and Z2 nontrivial semidirect product of Z4 and Z4
nontrivial 2 direct product of Z4 and Z2 SmallGroup(16,3)

Note that in \operatorname{Ext}^1, all the three nontrivial elements are in the same orbit under the natural action of the automorphism group of the Klein four-group. But they split into two orbits when we consider those automorphisms of the Klein four-group that are induced by automorphisms of the dihedral group:D8, which is why they give different outputs in H^2(G;A).

Homomorphisms to and from other cohomology groups

Homomorphisms on A

The unique injective homomorphism A = \mathbb{Z}_2 to \mathbb{Z}_4 induces a homomorphism:

\! H^2(G;A) \to H^2(G;\mathbb{Z}_4)

The group on the right is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of D8 on Z4). However, the induced map above is not an isomorphism. Rather, it has kernel of order four precisely the image of \operatorname{Ext}^1(G^{\operatorname{ab}},A) in H^2(G;A) (see the direct sum decomposition section) and its image is a subgroup of order two in H^2(G;\mathbb{Z}_4).

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 \mathbb{Z}_2 in \mathbb{Z}_4.

The map is given in the table below:

Input Number of copies Output = central product of input group with \mathbb{Z}_4 over identified central subgroup \mathbb{Z}_2
direct product of D8 and Z2 1 direct product of D8 and Z4
nontrivial semidirect product of Z4 and Z4 1 direct product of D8 and Z4
SmallGroup(16,3) 2 direct product of D8 and Z4
dihedral group:D16 1 central product of D16 and Z4
generalized quaternion group:Q16 1 central product of D16 and Z4
semidihedral group:SD16 2 central product of D16 and Z4

The unique surjective map from \mathbb{Z}_4 to A = \mathbb{Z}_2 induces a homomorphism:

H^2(G;\mathbb{Z}_4) \to H^2(G;A)

The kernel of this map is the image of the preceding map and the image of this map is the kernel of the preceding map. The map is given in the table below:

Input Number of copies Output
direct product of D8 and Z4 1 direct product of D8 and Z2
nontrivial semidirect product of Z4 and Z8 1 nontrivial semidirect product of Z4 and Z4
SmallGroup(32,5) 2 SmallGroup(16,3)
central product of D16 and Z4 1 direct product of D8 and Z2
SmallGroup(32,15) 1 nontrivial semidirect product of Z4 and Z4
wreath product of Z4 and Z2 2 SmallGroup(16,3)

GAP implementation

Construction of the cohomology group

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

gap> G := DihedralGroup(8);;
gap> A := TrivialGModule(G,GF(2));;
gap> T := TwoCohomology(G,A);
rec( group := <pc group of size 8 with 3 generators>,
  module := rec( field := GF(2), isMTXModule := true, dimension := 1,
      generators := [ <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2>,
          <an immutable 1x1 matrix over GF2> ] ),
  collector := rec( relators := [ [ 0 ], [ [ 2, 1, 3, 1 ], [ 3, 1 ] ],
          [ [ 3, 1 ], [ 3, 1 ], 0 ] ], orders := [ 2, 2, 2 ],
      wstack := [ [ 1, 1 ], [ 2, 1, 3, 1 ] ], estack := [  ],
      pstack := [ 3, 5 ], cstack := [ 1, 1 ], mstack := [ 0, 0 ],
      list := [ 0, 0, 0 ],
      module := [ <an immutable 1x1 matrix over GF2>, <an immutable 1x
            1 matrix over GF2>, <an immutable 1x1 matrix over GF2> ],
      mone := <an immutable 1x1 matrix over GF2>,
      mzero := <an immutable 1x1 matrix over GF2>, avoid := [  ],
      unavoidable := [ 1, 2, 3, 4, 5, 6 ] ),
  cohom := <linear mapping by matrix, <vector space of dimension 4 over GF(
    2)> -> ( GF(2)^3 )>,
  presentation := rec( group := <free group on the generators [ f1, f2, f3 ]>
        ,
      relators := [ f1^2, f1^-1*f2*f1*f3^-1*f2^-1, f2^2*f3^-1, f1^-1*f3*f1*f3\
^-1, f2^-1*f3*f2*f3^-1, f3^2 ] ) )

Construction of extensions

The extensions can be constructed using the additional command Extensions.

gap> G := DihedralGroup(8);;
gap> A := TrivialGModule(G,GF(2));;
gap> L := Extensions(G,A);;
gap> List(L,IdGroup);
[ [ 16, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 8 ],
  [ 16, 3 ], [ 16, 9 ] ]

Under the action of the various automorphism groups

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);
[ [ 16, 11 ], [ 16, 8 ], [ 16, 3 ], [ 16, 7 ], [ 16, 4 ], [ 16, 9 ] ]