Group extension problem

Statement
The group extension problem for two groups $$N$$ and $$Q$$ is the problem of finding all groups $$G$$ with $$N$$ as a normal subgroup of $$G$$, and the quotient group $$G/N$$ isomorphic to $$Q$$.

Congruence classes formulation
In this formulation, we're thinking of $$N$$ and $$Q$$ as specific groups, and looking at short exact sequences:

$$1 \to N \to G \to Q \to 1$$

where two short exact sequences:

$$1 \to N \to G_1 \to Q \to 1$$

and:

$$1 \to N \to G_2 \to Q \to 1$$

are equivalent if there is an isomorphism from $$G_1$$ to $$G_2$$ that induces the identity map on $$N$$ and on $$Q$$.

Group extensions that are equivalent in this fashion are called congruent group extensions and the equivalence classes are called congruence classes of group extensions.

Formulation upto automorphisms, or "pseudo-congruence"
This is a more general formulation, where we declare two short exact sequences:

$$1 \to N \to G_1 \to Q \to 1$$

and:

$$1 \to N \to G_2 \to Q \to 1$$

are equivalent if there is an isomorphism from $$G_1$$ to $$G_2$$ that induces an automorphism on $$N$$ and an automorphism on $$Q$$.

Group extensions that are equivalent in this fashion are called pseudo-congruent group extensions and the equivalence classes are called pseudo-congruence classes of group extensions.

Classifying group extensions for an abelian normal subgroup
If $$N$$ is an abelian group, then the following is a procedure to classify all group extensions with normal subgroup $$N$$.

Finding a list of possible actions
The first step is to note that the quotient group acts on the normal subgroup. In other words, given any group $$G$$ with a specified normal subgroup $$N$$ and a quotient group $$Q$$, there is a homomorphism:

$$\varphi:Q \to \operatorname{Aut}(N)$$

This automorphism is determined by the congruence class of the extensions, so we can partition all congruence classes of extensions as a disjoint union of congruence classes of extensions corresponding to elements of $$\operatorname{Hom}(Q,\operatorname{Aut}(N))$$.

Identifying homomorphism types for pseudo-congruence
If we're looking at extensions modulo equivalence up to automorphisms (i.e., pseudo-congruence), then an extension doesn't define a unique map $$\varphi:Q \to \operatorname{Aut}(N)$$. Rather, the map is defined uniquely up to pre-composition with automorphisms of $$Q$$ and conjugation by automorphisms of $$N$$. Thus, when classifying equivalence classes of extensions in this fashion, it suffices to consider equivalence classes of elements in $$\operatorname{Hom}(Q,\operatorname{Aut}(N))$$ under these actions, or rather, under the combined action of $$\operatorname{Aut}(N) \times \operatorname{Aut}(Q)$$.

Finding all congruence classes for a given action
the next step is to find all congruence classes of extensions that give rise to this homomorphism. This is, in fact, described using the second cohomology group $$H^2_\varphi(Q;N)$$. There are a number of shortcuts to computing this group when $$Q$$ or $$N$$ have special structure (for instance second cohomology group for trivial group action of finite cyclic group on finite cyclic group).

Finding all pseudo-congruence classes for a given action
Multiple congruence classes of extensions may again be equivalent up to automorphisms (i.e., they may be pseudo-congruent). The set of pseudo-congruence classes of extensions for a given action can be obtained from the set of congruence classes as follows: take the equivalence classes under the natural action of the following subgroup of $$\operatorname{Aut}(N) \times \operatorname{Aut}(Q)$$: $$C_{\operatorname{Aut}(N)}(\varphi(Q)) \times W$$ where $$W$$ is the subgroup of $$\operatorname{Aut}(Q)$$ comprising those automorphisms that do not affect the image under $$\varphi$$, i.e., $$W = \{ \sigma \in \operatorname{Aut}(Q) \mid \varphi(\sigma(q)) = \varphi(q) \ \forall \ q \in Q \}$$.

Note in particular that if $$\varphi$$ is trivial, then the acting group is the entire group $$\operatorname{Aut}(N) \times \operatorname{Aut}(Q)$$.

Classifying group extensions for an arbitrary normal subgroup
As before, we denote the normal subgroup as $$N$$ and the quotient group as $$Q$$.

Finding a list of outer actions
The first step is to note that quotient group maps to outer automorphism group of normal subgroup. Thus, for any group extension with normal subgroup identified with $$N$$ and quotient group identified with $$Q$$, we can construct a homomorphism:

$$\alpha:Q \to \operatorname{Out}(N)$$

where the group on the right is the outer automorphism group of $$N$$. This homomorphism is well-defined for group extensions up to congruence, i.e., congruent group extensions define identical homomorphisms.

In the case that $$N$$ is abelian, this reduces to the situation discussed earlier, because in that case the outer automorphism group is identified with the automorphism group.

Thus, a first step to classifying all the group extensions with normal subgroup $$N$$ and quotient group $$Q$$ is to determine the set $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$ of all possible group homomorphisms from $$Q$$ to $$\operatorname{Out}(N)$$. Then, for each such homomorphism, we will determine all the possible group extensions whose corresponding homomorphism is that homomorphism. The overall set of congruence classes of group extensions is thus a disjoint union over $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$ of the set of group extensions for each element therein.

Identifying homomorphism types for pseudo-congruence
If we are looking at extensions modulo automorphisms, i.e., up to pseudo-congruence, then it is no longer the case that we get a unique homomorphism from $$Q$$ to $$\operatorname{Out}(N)$$. Rather, we need to consider the orbits in $$\operatorname{Hom}(Q,\operatorname{Out}(N))$$ under the natural action of $$\operatorname{Aut}(N) \times \operatorname{Aut}(Q)$$ where $$\operatorname{Aut}(N)$$ acts by conjugation on the output (if we want, we can first quotient to $$\operatorname{Out}(N)$$ and then act by conjugation) and $$\operatorname{Aut}(Q)$$ acts by pre-composition. For pseudo-congruence classes, thus, it suffices to identify the pseudo-congruence classes of extensions for each such orbit.

Finding all congruence classes for a given outer action
There is a canonical homomorphism associated with the group $$N$$ as an abstract group (see outer automorphism group maps to automorphism group of center):

$$\pi: \operatorname{Out}(N) \to \operatorname{Aut}(Z(N))$$

where $$Z(N)$$ denotes the center of $$N$$.

We define the composite homomorphism $$\varphi = \pi \circ \alpha$$. Thus, $$\varphi \in \operatorname{Hom}(Q,\operatorname{Aut}(Z(N))$$. Now, consider the group $$H^2_\varphi(Q,Z(N))$$.

First, we note that the group $$H^2_\varphi(Q,Z(N))$$ acts canonically on the set of group extensions with normal subgroup $$N$$, quotient group $$Q$$, and outer action $$\alpha$$.

Our claim is that there are two possibilities:


 * 1) There are no group extensions with normal subgroup $$N$$, quotient group $$Q$$, and outer action $$\alpha$$.
 * 2) The canonical action of $$H^2_\varphi(Q,Z(N))$$ on the set of group extensions with normal subgroup $$N$$, quotient group $$Q$$, and outer action $$\alpha$$ is equivalent to the regular group action, i.e., it is transitive with a single orbit. In particular, if we "fix a basepoint" we can get a bijection from $$H^2_\varphi(Q,Z(N))$$ to the set of extensions.

The third cohomology group element criterion
To figure out whether case (1) or case (2) is operative, we need to compute a particular element of the third cohomology group $$H^3_\varphi(Q,Z(N))$$. If this element is a non-identity element of the group, then case (1) holds. If this element is the identity element of the group, then case (2) holds. Note in particular that if the third cohomology group is trivial, then case (2) must hold, and we do not need to compute the element explicitly in that case.

Here is how the element is computed.

The absence of canonical basepoint
As noted, in case (2), i.e., in the case that there do exist extensions corresponding to a particular outer action, the group $$H^2_\varphi(Q,Z(N))$$ acts on the set of congruence classes of extensions in a manner equivalent to the regular group action. In particular, if we choose a specific congruence class of extension as "basepoint" then we can get a bijection from $$H^2_\varphi(Q,Z(N))$$ to the set of extensions.

Recall that in the case that $$N$$ is an abelian normal subgroup, there is a canonical choice of basepoint, namely, the external semidirect product $$N \rtimes Q$$ corresponding to the action. However, in the general case, there need not be a canonical choice of basepoint. The automorphism action structure may preclude such a possibility. In particular:


 * There may not be any extension that is a semidirect product.
 * There may be more than one congruence class of extension that is a semidirect product. For instance, in the extensions for nontrivial outer action of Z2 on D8, both the extensions can be viewed as semidirect products. The same is true for both the extensions for trivial outer action of Z2 on D8, although in the trivial outer action case, we might legitimately consider the direct product the canonical basepoint.

The relationship between the outer action and the action on the center
The classifying invariant we use is the outer action $$\alpha:Q \to \operatorname{Out}(N)$$. On the other hand, the object (the second cohomology group) that classifies the set of extensions for a given $$\alpha$$ depends on something weaker than $$\alpha$$, the homomorphism $$\varphi:Q \to \operatorname{Aut}(Z(N))$$ obtained as $$\pi \circ \alpha$$.

It often happens that there are multiple choices of $$\alpha$$ that correspond to the same $$\varphi$$. In this case, we get a full separate set of congruence classes for each different $$\alpha$$ (or rather, for those $$\alpha$$s for which the set is non-empty). For instance, both the trivial and the nontrivial outer action of cyclic group:Z2 on dihedral group:D8 induce a trivial group action of cyclic group:Z2 on the center of dihedral group:D8. Thus, each of them gives an extension set that is a copy of the second cohomology group for trivial group action of Z2 on Z2, and these are distinct copies (in this case, the extensions in each copy are not even equivalent up to pseudo-congruence); see extensions for trivial outer action of Z2 on D8 and extensions for nontrivial outer action of Z2 on D8.