group extension properties: split, central, cocycle or not

Question

Given a group extension $$0 \to A \to E \to G \to 1 $$ with $A$ an abelian group, there are several properties to describe this extension:

1. central or non-central extension

2. split or not

3. trivial or nontrivial cocycle $H^2(G,A)$

4. direct product $A \times G$ or semi-direct product $A \rtimes G$

5. trivial or nontrivial map from $G \to Aut(A)$

Question -for each of properties from 1,2,3,4,5 above, we can either have positive (nontrivial) or negative (trivial) for the five descriptions. Namely, we have $2^5$ possibilities at most to use the above descriptions to describe the total $E$. My concern is that are all 5 options are all independent to each other? Or can it be that some of the properties determine the other properties (they are non indepdent?)

Possible useful ref: https://terrytao.wordpress.com/2010/01/23/some-notes-on-group-extensions/#more-3383

Answer 1

This is an odd way to slice things up.

1. For the extension to be either a direct or a semidirect product requires that it be split, so 2 and 4 aren't independent.
2. For the extension to be a central extension requires that the map $G \to \text{Aut}(A)$ be trivial, so 1 and 5 aren't independent either.
3. The cocycle in $H^2(G, A)$ classifying the extension is trivial iff the extension splits and is a semidirect product, so 3 and 4 aren't independent either.

The classification goes like this. The conjugation action of $E$ on $A$ induces an action $G \to \text{Aut}(A)$ (this step requires that $A$ is abelian), and then fixing such an action the possible extensions are classified by $H^2(G, A)$. The trivial cocycle corresponds to the semidirect product $A \rtimes G$. This means we can split things up into $4$ cases (not $32$), given by your properties 3 and 5:

1. Trivial action, trivial cocycle. Here $E = A \times G$ is a trivial semidirect product, or equivalently a trivial central extension, with the obvious inclusion and projection. For example, we can take $A = G = C_2, E = C_2 \times C_2$.
2. Trivial action, nontrivial cocycle. Here $E$ is a central extension. For example, we can take $A = G = C_2, E = C_4$.
3. Nontrivial action, trivial cocycle. Here $E = A \rtimes G$ is a semidirect product. For example, we can take $A = C_3, G = C_2, E = S_3 \cong D_3$.
4. Nontrivial action, nontrivial cocycle. Here I have to admit I don't know an easy small example. But here's a family of examples: consider the extension $1 \to N \to SL_2(\mathbb{Z}/p^2) \to SL_2(\mathbb{Z}/p) \to 1$ where $N = 1 + p M_2(\mathbb{Z}/p^2) \cong (\mathbb{Z}/p)^4$. $N$ is abelian but not central, and the extension doesn't split (the unipotent element $\left[ \begin{array}{cc} 1 & 1 \\ 0 & 1 \end{array} \right] \in SL_2(\mathbb{Z}/p)$ can't lift to an element of order $p$ although the proof I have is a slightly annoying computation), so it's neither a central extension nor a semidirect product. The action of $SL_2(\mathbb{Z}/p)$ on $N$ is the adjoint representation on $\mathfrak{sl}_2(\mathbb{Z}/p)$ but I don't know much about the cocycle.