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How do construct a group with a specified lower central series? Let's try the example where the series terminates at the integers:

$$ G = G_1 \unrhd G_2 \unrhd G_3 = \mathbb{Z} $$

What kind of number systems could these be? Could these be matrix-groups? Are they necessarily of the matrix type.

Here is one example already, upper-triangular matrices:

$$ \left[ \begin{array}{cccc} 1 & \mathbb{Z} & \mathbb{Z} & \mathbb{Z} \\ 0 & 1 & \mathbb{Z} & \mathbb{Z} \\ 0 & 0 & 1 & \mathbb{Z} \\ 0 & 0 & 0 & 1\end{array} \right] $$

One possible source of algebra problems like this could be Dummit & Foote. Or maybe on arXiv.

Basically, how many 3-step nilpotent groups are there? We have commutator conditions $[G,G] = [G_1, G_1] \subseteq G_2$ and $[G_1, G_2] \subseteq G_3 = \mathbb{Z}$.

Example: $G = \mathbb{Z}^3 = \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z}$ and then $[G, G] = \{ e\} \subseteq \mathbb{Z}$. This group is commutative.

We could try to construct group with generators $\{ e_1, e_2, e_3\}$ and give some kind of commutator relation of the elements $[e_i, e_j] = ...$ not sure how to do that. Or how these match up with the matrix example.

Comments suggest the term "2-cocycle" learning what this term might mean.

Also discussion from Physics Page

cactus314
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  • Do you know how to construct central extensions from $2$-cocycles? – Qiaochu Yuan Jan 14 '21 at 18:34
  • I am not really sure what you are asking. You have asked a lot of questions, but they sound more like general speculation rather than precise questions that would admit an answer. – Derek Holt Jan 14 '21 at 19:02
  • @QiaochuYuan undergrad in abstract algebra might not know what "2-cocycle" is. There's interesting note "2-cocycle of trivial group action" get's a separate page. – cactus314 Jan 14 '21 at 19:24
  • You can get further examples by taking a direct product of your example with any nilpotent group of class at most two. – Derek Holt Jan 17 '21 at 20:26

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