# group extension properties: split, central, cocycle or not

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?)

• Split extensions are semidirect products. Direct products are semiderect too. So there are fewer than $2^5$ possibilities. Sep 27, 2020 at 6:12

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.
• Thank you! @Qiaochu Yuan, does this example also fit into your answer? math.stackexchange.com/questions/135444/… Sep 27, 2020 at 13:59
• @annie: any abelian extension of abelian groups is a central extension. Whether $E$ is abstractly isomorphic to $A \times G$ is a different question from whether $E$ is the trivial extension. Sep 27, 2020 at 17:20
• thanks I will accept as an answer in 7 days. Sep 27, 2020 at 19:26
• do you happen to know this math.stackexchange.com/questions/3842714/ ? Sep 27, 2020 at 19:27
• How could we reproduce the group multiplication rule of 𝑔1⋅𝑔2∈𝑆𝑝𝑖𝑛(𝑑) via the data of ℤ/2, 𝑆𝑂(𝑑),𝑓=𝑤2(𝑉𝑆𝑂(𝑑))∈𝐻2(𝐵𝑆𝑂(𝑑),ℤ/2)=ℤ/2,𝑟:𝑆𝑂(𝑑)→𝐴𝑢𝑡(ℤ/2) there as an example. Sep 27, 2020 at 19:29