# 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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### Quotient of a non-elementary nilponent group

Here is a (seemingly) simple problem in group theory. Given a non-elementary finite nilpotent group $N$, show there exist $p \neq q$ primes such that $N$ has a quotient $\Bbb Z_{pq}^{2}$. Here, an ...
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### Sufficient condition for nipotency of a Lie group

Let $G$ be a Lie group such that its Lie algebra $\mathfrak{g}$ admits a decomposition of the form $$\mathfrak{g}=\Delta\oplus [\mathfrak{g},\mathfrak{g}].$$ Where $\Delta$ is a bracket generating ...
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### Non-solvable Lie Algebra [duplicate]

I read that $\mathfrak{sl}_2(\mathbb{C})$, the set of $2 \times 2$ matrices over $\mathbb{C}$, with trace zero, is not solvable. Could someone please explain or show why this is the case? I cannot ...
1 vote
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### Dummit and Foote 6.1.2

Let $P$ be a finite nilpotent group. We want to show that every proper subgroup of $P$ is a proper subgroup of its normalizer in $P$. I was following the solution. But there are certain parts I didn't ...
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### Let $G_1=G/Z(G)$ for a group $G$. Let $G_{i}$ be the group $G_{i-1}/Z(G_{i-1})$. If $G_{k}/Z(G_{k})$ is abelian, is $G$ nilpotent? [closed]

Let $G$ be a group and $Z(G)$ be its center. Now let $G_{1}$ be the group $G/Z(G)$ and $Z(G_{1})$ be its center. Inductively let $G_{i}$ be the group $G_{i-1}/Z(G_{i-1})$. If for some $k$ we have ...
1 vote
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### Let $G$ be a nilpotent group and $|G| = p_1p_2p_3$ where $p_i$ are different prime numbers. Prove that $G$ is an abelian group.

Let $G$ be a nilpotent group and $|G| = p_1p_2p_3$ where $p_i$ are different prime numbers. Prove that $G$ is an abelian group. It has been a long time since I worked with algebra and I am at a loss ...
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### Constructing central series

Problem Statement: Prove nilpotent group is supersolvable. A finite nilpotent group has supersolvable series The link above uses $p$-group and direct product. But suppose we do not do that, the ...
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### How to use the upper central series instead of the lower one in nilpotency proofs

I have recently been doing many exercises on nilpotent groups and I'm having a bit of a problem using the upper central series to prove theorems. I can prove them using the lower one just fine (mainly ...
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### Minimal normal subgroup is contained in Centre of group for nilpotent groups [duplicate]

This question is from Hungerford Algebra Chapter Structure of groups. If $G$ is a finite nilpotent group, then every minimal normal subgroup of $G$ is contained in $C(G)$ and has prime order. A ...
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### Nilpotent Group and Normal Series

I tried some questions in abstract algebra (Groups, Rings, Modules, Fields) in January and February and I was unable to solve some. I am asking those now because I was critically ill then. I have done ...
1 vote
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### Solvability, nilpotence and permutation Groups

I had tried some questions in Group Theory in January but Could not post the questions on which I am struck because of my illness. So, I am writing them now. I have done a graduate level course on ...
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### If $|H| = p$ s.t. $H<N<S_p$ where $N$ is nilpotent, then $H = N$. [duplicate]

Given $p$ prime and $H \le S_p$ such that $|H| = p$, show that if $H \leq N \leq S_p$ for some nilpotent subgroup $N$, then $H=N$. The question was already asked. But I didn't understand the hint. So ...
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### Example of nilpotent profinite group

Is there any example of a topologically finitely generated profinite group $G$ which is nilpotent as a group but has an infinite torsion-part? The question is motivated by the fact that an algebraic ...
1 vote
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### Equivalent definition of nilpotency for infinite groups

Let $G$ be a finite group. TFAE: (1) $G$ is nilpotent. (2) $N_{G}(H) > H$ for all subgroups $H< G$. (3) Every maximal subgroup of $G$ is a normal subgroup. The proof of these equivalences seems ...
Let $N$ be a finitely generated infinite nilpotent group and let us denote by $G$ the semidirect product $N \rtimes \mathbb{Z}^n$ for some $n\in \mathbb{N}$. I would like to know if there is an ...