# 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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### Prove that the group of unipotent matrices is nilpotent. [duplicate]

I'm trying to prove that the group of unipotent matrices (upper triangular with $1$'s on the diagonal) is nilpotent, so that the lower central series terminates in the trivial group after finite ...
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### Proof of a particular piece of Milnor-Wolf theorem

The Milnor-Wolf theorem says that a finitely generated solvable group that doesn't have exponential growth is virtually nilpotent. The proof I've seen is divided into two pieces: Prove that such a ...
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### Examples of nilpotent subgroups of $S_n$ which have less than say 10^9 elements?

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 ...
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### Why is the direct product of finitely many nilpotent groups nilpotent? [duplicate]

I want to ask a question, and I found it here: Why is the direct product of a finite number of nilpotent groups nilpotent? But I am struggling to understand how can we take the product of two normal ...
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### Is there an infinite nilpotent group with one Sylow subgroup that is not normal? [closed]

It is known that a finite nilpotent group has every Sylow subgroup normal in it. Does this result generalize to infinite nilpotent groups or not ? If not, why, what is a counter example?
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### Finitely generated, nilpotent, torsion-free group that is also radicable

I am currently working with Mal'cev completions, using the following definition: Let $N$ be group that is Nilpotent Torsion-free Finitely generated Then the Mal'cev completion or radicable hull is ...
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### If $G$ is nilpotent then so is $G/Z(G)$.

The Problem: If $G$ is nilpotent then so is $G/Z(G)$. My Background: Chapter 1-5 of Abstract Algebra $\mathit{3^{rd}}$ edition by Dummit and Foote. For any group $G$ define the following subgroups ...
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### All maximal subgroups are Sylow subgroups

Let $G$ be a group in which all maximal non-trivial subgroups are Sylow subgroups. Then $G$ isn't a simple non-abelian group. I know how to prove this by relying on the theorem that if all maximal ...
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### Alternative proof that maximal abelian normal subgroups of a nilpotent group are self-centralizing

I was writing lecture notes on nilpotent groups, and wanted to show the following well-known basic result. Theorem: Let $G$ be a nilpotent group and let $A$ be an abelian normal subgroup of $G$, ...
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### Nilradical in an Artinian ring

I am reading a book that suggests to find what $N(R)$ is - the nilradical of $R$ where $R$ is Artinian you just need to find a nilpotent ideal $I$ of $R$ and then show that $R/I$ has no nonzero ...
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### Nilpotent ideals in a ring proof

I am trying to understand a proof I am reading but it doesn’t make much sense to me. If $R$ is a ring with $I$ a nilpotent left ideal then $I$ is contained in a nilpotent two-sided ideal. The proof ...
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### Nilpotent of subgroup implies nilpotent of group

Suppose that I have two normal subgroups $L, M$ in $H$ such that $L<M$ and $M / L$ is nilpotent of class $2$. Suppose also that $[H,[H,H]]$ is contained in $[M,M]$. Can I then conclude that $H / L$ ...
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### Finite group where product of elements of coprime order has product order is nilpotent

Let $G$ be a finite group where for all $a, b \in G$ with $\operatorname{gcd}(o(a), o(b)) = 1$, $o(ab) = o(a)o(b)$. Is $G$ nilpotent? My try: Let's work by induction on $\lvert G \rvert$. Since every ...
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### Classification of nilpotent groups via $p$-groups

First some background: Some time ago I learned that since any finite abelian group is a direct product of (finite) cyclic groups, the finite cyclic groups ($\mathbb{Z}_n$) are the key to understanding ...
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### Proof of Baer's theorem

Baer's Theorem: Let $x$ be a $p$-element of a finite group $G$. Suppose that $\langle x,x^g\rangle$ is a $p$-subgroup for every $g\in G$. Then $x\in O_p(G)$. Here, $O_p(G)$ denotes the largest normal ...
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### Let $N \unlhd G$. Then $G/N$ is nilpotent of class $c\in\Bbb{N}$ iff $c$ is the smallest natural number such that $\gamma_c(G) \subset N$

I think I have established the following proposition: Theorem: Let $G$ be a group and $N \unlhd G$. Then, $G/N$ is nilpotent of class $c \in \mathbb{N}$ if, and only if, $c$ is the smallest natural ...
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### Property of Fitting subgroup: Let $G$ be a finite group and $C:= C_G(F(G))$. Then $O_p(C/C\cap F(G))=1$ for every prime $p$.

I'm trying to understand the proof of the following which is stated in Kurzweil and Stellmacher: Let $G$ be a finite group and $C:= C_G(F(G))$. Then $$O_p(C/C\cap F(G))=1$$ for every prime $p$. Here,...
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Can someone please check my proof of the following "evident" equivalence stated in Kurzweil and Stellmacher. A finite group $G$ is nilpotent if and only if $$U<N(U) \text{ whenever } U&... • 1,599 3 votes 1 answer 113 views ### How do subgroups of the inner automorphisms of a group look like? I'm trying to prove the following proposition: Let G be a group. Then G is nilpotent iff {\rm Inn}(G) is nilpotent. I've proven that if G is nilpotent then {\rm Inn}(G) is nilpotent as ... • 149 0 votes 2 answers 220 views ### Is it true that nilpotent always has some eigenvalue? I understand that if a nilpotent matrix has some \lambda eigenvector, then it implies that \lambda=0 because if$$Ax = \lambda x \\ A^2x = \lambda^2x \\ A^3x=\lambda^3x \\ \vdots \\ 0=A^k=\lambda^...
We know that these two conditions are equivalent for finite groups : i) $G$ is solvable. ii) If $\left|G\right|=mn\;$ and $(m,n)=1$ then G has a subgroup of order $m$. Now are the following two ...