Second cohomology group for trivial group action of D8 on Z4

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:Z4. The elements of this classify the group extensions with cyclic group:Z4 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:Z4|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 dihedral group:D8 on cyclic group:Z4, i.e., the group $\! H^2(G;A)$

where $G \cong D_8$ is dihedral group:D8 (order 8, degree 4) and $A \cong \mathbb{Z}_4$ is cyclic group:Z4.

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

Elements

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.

Note that all these extensions are central extensions with the base normal subgroup isomorphic to cyclic group:Z4 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 $4 \times 8 = 32$.

None of the extensions is a stem extension, because the Schur multiplier of dihedral group:D8 is cyclic group:Z2, which does not admit cyclic group:Z4 as a quotient.

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.

Cohomology class type Number of cohomology classes Corresponding group extension Second part of GAP ID (order is 32) 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 Z4 25 No No $\Gamma_2$ 2 2 3
nontrivial and symmetric 1 nontrivial semidirect product of Z4 and Z8 12 No Yes $\Gamma_2$ 2 2 2 center of nontrivial semidirect product of Z4 and Z8
nontrivial and symmetric 2 SmallGroup(32,5) 5 No No $\Gamma_2$ 2 2 2
nontrivial 1 central product of D16 and Z4 42 No Yes $\Gamma_3$ 3 2 2 center of central product of D16 and Z4
non-symmetric 1 SmallGroup(32,15) 15 No Yes $\Gamma_3$ 3 2 2 center of SmallGroup(32,15)
non-symmetric 2 wreath product of Z4 and Z2 11 No Yes $\Gamma_3$ 3 2 2 center of wreath product of Z4 and Z2
Total (6 rows) 8 -- -- -- -- $\Gamma_2, \Gamma_3$ -- -- -- --

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. Under this action, there are four fixed points: direct product of D8 and Z4, nontrivial semidirect product of Z4 and Z8, central product of D16 and Z4, and SmallGroup(32,15). There are two orbits of size two, corresponding to the extensions SmallGroup(32,5) and wreath product of Z4 and Z2 respectively.

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

where $G^{\operatorname{ab}}$ 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. Also, $H_2(G;\mathbb{Z})$ 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 extensions for symmetric cohomology classes, and these are precisely the ones that yield groups that overall have nilpotency class two. It has two cosets in the whole second cohomology group, with its non-identity coset comprising precisely the extensions that have overall nilpotency class three.

A natural choice of complement is the subgroup that arises as the kernel of the homomorphism $H^2(G;A) \to H^2(G;\mathbb{Z}_2)$ induced by the natural surjective map from $A = \mathbb{Z}_4$ to $\mathbb{Z}_2$. This subgroup comprises precisely the trivial extension (direct product of D8 and Z4) and central product of D16 and Z4.

Below is a depiction of the elements of the second cohomology group, where the rows are cosets of the image of $\operatorname{Ext}^1$ and the columns are cosets of our chosen complement:

Homomorphisms to and from other cohomology groups

Homomorphisms on $A$

The unique injective homomorphism $\mathbb{Z}_2$ to $A = \mathbb{Z}_4$ induces a homomorphism: $\! H^2(G;\mathbb{Z}_2) \to H^2(G;A)$

The group on the left is also isomorphic to elementary abelian group:E8 (see second cohomology group for trivial group action of D8 on Z2). However, the induced map above is not an isomorphism. Rather, it has kernel of order four precisely the image of $\operatorname{Ext}^1(G^{\operatorname{ab}},\mathbb{Z}_2)$ in $H^2(G;\mathbb{Z}_2)$ (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 $A = \mathbb{Z}_4$ to $\mathbb{Z}_2$ induces a homomorphism: $H^2(G;A) \to H^2(G;\mathbb{Z}_2)$

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)