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To make some computational experiments with finite nilpotent group - it would be helpful to know the following:

Question: What are the examples of nilpotent (but not commutative) subgroups in symmetric group $S_n$ which are not so big - say less than $10^9$ elements ?

Preferably given by explicit generators (permutations).

Alexander Chervov
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    Sylow subgroups of symmetric groups, or other permutation groups? You could take direct products of these to get non$p$-groups. – Derek Holt Feb 09 '24 at 14:51
  • @DerekHolt Thank you ! Are there explicit generators for Sylow subgroups ? (beyond S_p where they are cyclic ) PS products - not what I need - that are too simple. Sorry for not mentioning it in the question . – Alexander Chervov Feb 09 '24 at 15:12
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    Yes there are explicit generators. Since you are talking about computer experiments, I was assuming that you would be familiar with GAP or Magma. They include functions that return the subgroups with generators. – Derek Holt Feb 09 '24 at 16:50
  • I would prefer explicit permutations, since GAP seems cannot compute growth, but thank you any way ! – Alexander Chervov Feb 09 '24 at 17:05
  • Are you asking for (a) nilpotent groups of order $\leq 10^9$ or (b) nilpotent groups that embed in $S_n$, $|S_n| < 10^9$, i.e., nilpotent groups that have a faithful permutation representation on at most $12$ letters? – Travis Willse Feb 09 '24 at 18:17
  • @TravisWillse (a) – Alexander Chervov Feb 09 '24 at 18:22

2 Answers2

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The smallest non-abelian nilpotent group is the dihedral group of order 8 the symmetries of a square. This can be realized as a permutation group by keeping track of what a square symmetry does to vertices. Then a rotation acts on vertices by cycling them: $(1 \; 2\; 3 \; 4)$ and a reflection acts by $(1 \; 2)(3 \; 4)$. These generate a copy of $D_4$ in $S_4$.

If you want to do computations, here are two other sources of examples:

  1. Group extensions: a group is nilpotent iff it is an iterated extension by abelian groups. If you're comfortable with semi-direct products, these can lend themselves well to computations.

  2. Upper triangular matrices with 1s on the diagonal. In particular, the Heisenberg group. These groups lend themselves well to computations, and give good intuition about what nilpotent groups ``actually" are when you think about how they act on a vector space.

Noah Caplinger
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  • Thank you ! that I have considered already. I am interested in some other. up-triag of course main source - but soon running out of resources even on Z/2 – Alexander Chervov Feb 09 '24 at 14:09
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All $p$-groups, i.e., groups whose order is a prime power $p^m$, are nilpotent, so any nonabelian group of order such a power is an example (necessarily $m \geq 3$). Conversely, all finite nilpotent groups are direct products of groups of prime power orders.

So, the nilpotent, nonabelian groups of order $< 2^5$ are:

  • Order 8
    • dihedral group, $D_8 \cong \langle (14)(23), (1234) \rangle$
    • quaternion group, $Q_8 \cong \langle (1247)(3685), (1348)(2576) \rangle$
  • Order 16: all $9$ nonabelian groups
  • Order 24: $C_3 \times D_8$, $C_3 \times Q_8$
  • Order 27: $(C_3 \times C_3) \rightthreetimes C_3$, $C_9 \rightthreetimes C_3$

All of these groups can be realized as groups of permutations of $n \leq 12$ objects, except for $Q_{16}$, the generalized quaternion group, which embeds in no symmetric group smaller than $S_{16}$.

Travis Willse
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