# Questions tagged [nilpotent-groups]

A nilpotent group is: 1) a group, whose upper central series stabilizes after a finite length at the whole group. 2) a group, whose lower central series stabilizes after a finite length at the trivial subgroup. 3) a group, that possesses a central series. All those definitions are equivalent. To be used with the tag [group-theory].

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### An explicit representation of a free nilpotent group by unitriangular matrices

P. Hall proved that every finitely generated torsion-free nilpotent group can be faithfully represented by upper unitriangular matrices over $\mathbb Z$. The most famous example is the integral ...
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### How to find minimal generating sets efficiently?

Let $G$ be a group. A minimal generating set of $G$ is a subset of $R$ of $G$ that generates $G$ such that no proper subset of $R$ also generates $G$. For arbitrary groups is its not true (unlike ...
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### Show that these two definitions of a nilpotent group are equivalent.

A group $G$ is nilpotent if there exists of a normal series $1 = G_0 \subseteq G_1 \subseteq G_2 \subseteq \dots \subseteq G_n = G$ such that $G_{i + 1}/G_i \subseteq Z(G/G_i)$. But now I'm seeing ...
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### Let $G$ a nilpotent group such that $10$ divides $|G|$. It is true that $G$ has element of order $10$?

Let $G$ a nilpotent group such that $10$ divides $|G|$. It is true that $G$ has element of order $10$? We know that $G$ can be expressed like direct sum of Sylow's subgroups and that $G$ has a normal ...
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### Is every simply-connected nilpotent Lie group algebraic?

Say a Lie group is a matrix group if it is a closed Lie subgroup of some general linear group. Say a Lie group is algebraic if it is the group of $\mathbb R$-points of a real linear algebraic group. ...
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### Number of conjugacy classes of a group of order $5^4$ whose center is $25$ order

How many conjugacy classes does a group of order $625$ have, if its center is of order $25$? I know that the orders of conjugacy classes should divide the order of the group, so this leaves with each ...
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### Nilpotent groups can be constructed by means of abelian groups.

I am studying A course in theory of groups by Robinson. When defining the central extension of groups, the author says that Every nilpotent group can be constructed from abelian groups by means of a ...
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### Having trouble understanding nilpotent and supersolvable group in terms of short exact sequence

$G$ is supersolvable if $G_{i+1}/G_i$ is cyclic for every $i\geq 0$ where $G_i$ is a normal sequence of $G=G_n$. According to the lecture note I read, it says that in the following short exact ...
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### Semidirect product is supersolvable groups

Let $G$ be a finitely generated group and $N$ be a nilpotent group (not necessarily finitely generated) such that $N$ is a subgroup of $G$ and $G / N$ is $\mathbb{Z}$. That is, there is a short exact ...
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### Characterization of a virtually nilpotent group

Let $G$ be a group that has an increasing sequence of subgroups $G_i \le G_{i+1}$ satisfying the following properties. (a) $G=\bigcup_{i\ge 1} G_i$. (b) Each $G_i$ contains a nilpotent subgroup $N_i$ ...
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### order of trace character distribution of irreducible unitary representations of nilpotent groups.

Let $N$ be a simply connected nilpotent group and let $\pi$ be an irreducible unitary representation of $N$. Then for every Schwartzfunction $f \in S(N)$ the integrated Operator $\pi(f)$ is of trace ...
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### nilpotent uniform pro-$p$ groups of dimension 2

I read in a paper that all nilpotent uniform pro-$p$ groups of dimension $\leq 2$ are Abelian (Prime Decomposition and the Iwasawa $\mu$-Invariant by Hajir-Maire, Math. Proc. of the Camb. Phil. Soc. (...
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### Do finitely generated nilpotent groups contain torsion free subgroups of finite index?

I have a question about the proof of proposition 6.9 of the paper "Rational Subgroups of Biautomatic Groups" by Gersten and Short (available here). The proposition states that a finitely presented ...
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### Ascending central series for dihedral group $D_4$ of order 8

Assume the dihedral group of order $2n$ is defined by $$D_n=⟨s,r∣r^n=s^2=e,srs^{-1}=r^{-1}⟩$$ I know that $D_n$ is nilpotent iff $n=2^m$ for some $m\ge0$. I am asked to directly show that $G=D_4$ is ...
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### Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$?

Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$? Here $\mathfrak{N}_2$ stands for the variety of nilpotent groups of class $2$ (defined by the identity $[[x, y], z]$). ...
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### If a group has a normal subgroup of order $d$ for every divisor $d$ of the order of the group then it is nilpotent

How do I prove this? Let $G$ be a finite group. For every $d$ dividing $|G|$, there is a normal subgroup of order $d$. Then $G$ is nilpotent. The proof in my notes says that clearly every Sylow ...
### Why isn't $Z_2 \times S_3$ nilpotent?
I have just learned the definition of a nilpotent group. My book seems to claim that $Z_2 \times S_3$ is not nilpotent, because they say, for the upper central series, $Z(G) = Z_1(G) = Z_2(G) = Z_n(G)$...