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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 posesses a central series. All those definitions are equivalent. To be used with tha tag [group-theory].

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Growth rate of free nilpotent group of rank $r$ and nilpotency index $s$.

Let $F^{(r)}$ denote the free group of rank $r$ with generators $x_1, \dots, x_r$. Recall a group $G$ is nilpotent of index $s$ if $G_{s+1} = \{e\} $ and $G_s \neq \{e\}$ (where $G_i$ denotes the ...
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97 views

Intersection of a nested sequence of subgroups

Let $G$ be a f.g. torsion-free nilpotent group and $(H_n)_{n\geq 1}$ a nested sequences of subgroups of finite index with trivial intersection. Question: Is it true that $\displaystyle\bigcap_{n\...
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1answer
21 views

Index of $G^n$ in $G$, an $s$-step nilpotent group of rank $\leq r$

For a group $G$ and $n \in \mathbb N$, let $G^n = \langle g^n \mid g \in G \rangle $ I am asked to show that if $G$ is $s$-step nilpotent and of rank at most $r$, then $[G:G^n] \leq n^{O_{r,s}(1)}$ ...
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1answer
38 views

If $G$ is $s$-step nilpotent and $n \in \mathbb N$, then $(G_s)^{n^s} \subset (G^n)_s \subset (G_s)^n$

Define the following "commutators" recursively: $[g,h] = g^{-1}h^{-1}gh$ $[g_1, \dots, g_m] = [g_1, [g_2, \dots, g_m]]$ for all $m \geq 3$ Let $G_i$ be the lower central series of a group, $G$. ...
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Finitely generated nilpotent group is isomorphic to a quotient of the free nilpotent group.

Let $F^{(r)}$ be the free group generated by $r$ elements. Let $\gamma_n(F^{(r)})$ denote its lower central series. Finally, let $F_{n,r} = F^{(r)}/\gamma_{n+1}(F^{(r)})$ be the free nilpotent group ...
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22 views

Growth rate of finitely generated nilpotent groups

Let $N$ be a group and $S$ a finite, symmetric generating set with the identity. For $n \in \mathbb N$, we let $S^n = \{s_1\dots s_n\mid s_i \in S\}$ We say $N$ has polynomial growth rate if $\...
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1answer
36 views

Let $G$ be a nilpotent group prove that for each $x \in Z_2(G)$ the map $\theta_x: G \rightarrow Z(G)$ defined by $\theta_x(g)=[g,x]$ is a hom

Let $G$ be a nilpotent group of class c. Prove that for each $x \in Z_2(G)$ the map $\theta_x: G \rightarrow Z(G)$ defined by $\theta_x(g)=[g,x]$ is a homomorphism with kernel $C_G(x)$. A hint is ...
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Existence of a nilpotent subgroup $N \leq G$ of step $\leq n$ such that a finite $A$ is in $K^{O_n(1)}$ left cosets of $N$

Some extra details left out of the title: Given a group $G$, a symmetric subset $A \subset G$ containing $1$ is called a $K$-approximate group if $|A^2| = |\{ab \mid a,b \in A\}| \leq K|A|$ We are ...
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1answer
61 views

Let $G$ be a $p$-group of nilpotency class at most 2, where $p$ is an odd prime. Then ${\{x^p| x \in G\}}$ is a subgroup of $G$.

Let $G$ be a $p$-group of nilpotency class at most 2, where $p$ is an odd prime.Then $A=\{{x^p| x \in G}\}$ is a subgroup of $G$. As $G$ is a group of nilpotency class at most 2, if the nilpotency ...
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1answer
20 views

A question about Frattini subgroup of specific form v2.0

Suppose $p$ is a prime number and $G$ is a finite group, such that $\Phi(G) = D_4 = \langle a \rangle_4 \rtimes \langle b \rangle_2$, where $\Phi$ denotes the Frattini subgroup. Is it always true, ...
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1answer
43 views

Why all nilpotent and finitely generated groups are Max?

An exercise asks to prove that if a group $G$ is nilpotent and finitely generated then it satisfy Max condition, in other words all non empty and totally ordered, with respect of inclusion $\subseteq$,...
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1answer
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Let $\zeta_{i}G$ be the upper central series of G. Why does the definition of central series imply that $H\zeta_{i}G$ is normal in $H\zeta_{i+1}G$.?

Let $\zeta_{i}G$ be the upper central series of G. Why does the definition of central series imply that $H\zeta_{i}G$ is normal in $H\zeta_{i+1}G$.? I am trying to show that a finite group $G$ being ...
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0answers
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Relationship between Carnot-Caratheory Distance and Levi-Civita Connection

Suppose that $G\cong H\times K$ is a nilpotent Lie group, where $H,K$ are Lie-Subgroups for which $H$ and $K$ are Lie-Subgroups such that $H$ is Commutative $K$ is Compact Then $G$ admits a bi-...
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1answer
119 views

A question about Frattini subgroup of specific form

Suppose $p$ is a prime number and $G$ is a finite group, such that $\Phi(G) = C_p \times C_p$, where $\Phi$ denotes the Frattini subgroup. Is it always true, that $p^4$ divides $|G|$? This statement ...
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1answer
36 views

$D_4 \times \mathbb{Z}_2$ different upper and lower central series

I wanted to find a group $G$ which had different upper and lower central series. Moreover, both different to one of his central series. I have found that $D_4\times \mathbb{Z}_2$ is one of the ...
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1answer
50 views

If $\operatorname{class}(G) = 2$ and $\exp(G) = 4$ then $\exp(G') = 2$?

Let $G$ be a finite $p$-group. I'd like to prove (or disprove) that if the nilpotency class of $G$ equals two (i.e., $1 \neq G' \le Z$, where $Z$ is the center of $G$) and the exponent of $G$ equals ...
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1answer
94 views

$G$ with a central series that is different from the upper and the lower central series

Let $$1=G_0\leq G_1\leq ... \leq G_{n-1} \leq G_n = G$$ be a central series of the group $G$. That is, $G_{i-1}/G_i\leq Z(G/G_i)$ for all $i$. Let $$1=Z_0(G)\leq Z_1(G)\leq ... $$ $$... \leq \...
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0answers
74 views

Finite Engel group is nilpotent.

A group $G$ is said to be $n$ engel if $$[x,[x, \dots ,[x,y]]\dots ]=1,$$ where $x$ appears $n$ times, and this holds for all $x,y\in G$. We know there is infinite order engel group which is not ...
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2answers
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Prove $H$ is a proper normal subgroup of $G$ if $H$ is generated by $\{[x,y] \mid x,y \in G\} \cup \{x^p \mid x \in G\}$.

I am trying to solve the following problem: Let $G$ be a nontrivial finite $p$-group, where $p$ is a prime, and let $H$ be the subgroup of $G$ generated by the set $\{[x,y] \mid x,y \in G\} \cup \{...
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0answers
43 views

Show that any finite nilpotent group of square free order is cyclic.

Show that any finite nilpotent group of square free order is cyclic. Hint: Suppose G is such a group. Any Sylow subgroup of G is of prime order. Hint: Any finite nilpotent group is the direct ...
5
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1answer
62 views

Let $G$ be a polycyclic group and assume that every finite quotient of $G$ is nilpotent. Then $G$ is nilpotent

Let $G$ be a polycyclic group and assume that every finite quotient of $G$ is nilpotent. Then $G$ is nilpotent. First some preliminaries: Every infinite polycyclic group contains a free ...
4
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1answer
85 views

Nilpotent ring and Nilpotent groups.

Let $R$ be a ring (associative and with unity) and $B$ be a subring with the property that $B^n = 0$ i.e. $$ \forall\; x_1, x_2, \dots, x_n \in B: \; x_1 \cdot x_2 \cdots x_n = 0$$ My aim is to ...
5
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1answer
69 views

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent.

Let $G$ be a finite group and $M,N \lhd G$ such that $M \leq N\cap \Phi(G)$. Then $\frac{N}{M}$ is nilpotent iff $N$ is nilpotent, where $\Phi(G)$ is the Frattini subgroup of $G$. The converse side ...
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1answer
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Connected Lie group for which every connected Lie subgroup is simply connected

Let $G$ be a simply connected Lie group. If $G$ is nilpotent, we know every connected Lie subgroup of $G$ is simply connected by Baker-Campbell-Hausdorff formula. What about the converse? If every ...
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0answers
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Let $Z_i(G)$ be the terms of the upper central series of $G$. Let $H \trianglelefteq G$. Show $Z_i(G)H/H \subseteq Z_i(G/H)$

Let $Z_0(G) = \{ 1 \}$ and: $$Z_{i + 1}(G)/Z_i(G) = Z(G/Z_i(G))$$ (as defined in dummit and foote). I want to show that $Z_i(G)H/H \subseteq Z_i(G/H)$. I can see it's true for $i = 0$ and $i = 1$, ...
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If $G/Z(G)$ is nilpotent then $G$ is nilpotent [duplicate]

Notation : $Z_{i+1}(G)=${$g\in G : [g,x]\in Z_i(G) \forall x\in G$} where $Z_0(G)=1$. Definition : $G$ is said to be nilpotent if $Z_n(G)=G$ for some $n\in \Bbb N$. Let $Z_n(G/Z(G))=G/Z(G)$. I have ...
2
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1answer
35 views

Use an alternative definition for nilpotent groups and show that $D_{8}$ is nilpotent.

Definition. The $i$-higher commutator subgroup $\mathcal{D}_{i}(G)$ is defined by $$\mathcal{D}_{1}(G) = G$$ and $$\mathcal{D}_{i+1}(G) = [\mathcal{D}_{i}(G),G].$$ A Course on Finite Groups. ...
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Let $G$ be a nilpotent group of class 3. Then for every $x ,y ,z$ in $G$, $[x,y,z][y,z,x][z,x,y]=1$.

Let $G$ be a nilpotent group of class 3. Then for every $x ,y ,z$ in $G$, $[x,y,z][y,z,x][z,x,y]=1$. As $G$ is a nilpotent group of class 3, $[G,G,G,G]=1$ and $G^{'}$ is abelian. I want to use Hall-...
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1answer
29 views

Locally compact nilpotent group has an open subgroup isomorphic to $\mathbb{R}^n\times K$

My question is about a possible generalization of the following structure theorem of locally compact abelian groups. Theorem: Let $G$ be a locally compact abelian group. Then here exists a compact ...
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2answers
71 views

When is the automorphism group of a finite $p$-group nilpotent?

Suppose $G$ is a finite $p$-group with odd $p$. Is it true, that $Aut(G)$ is nilpotent iff $G$ is cyclic? When $G$ is cyclic, $Aut(G)$ is indeed abelian and thus nilpotent. However, I do not know ...
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1answer
128 views

Solvable, non-nilpotent group with nilpotent commutator subgroup

What is the smallest example of a finite solvable, non-nilpotent group $G$, such that its derived subgroup $G'$ is nilpotent, but not abelian?
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Quotient of finite, nilpotent group by Frattini subgroup is isomorphic to product of quotients of Sylow subgroups by their respective Frattini groups.

Let $G$ be a finite nilpotent group. We know that $G=G_{p_1}\times G_{p_2}\times \cdots \times G_{p_r}$ where $G_{p_i}\in Syl_{p_i}(G)$, $i=1,\dots,r$. Is the following equation right? And why? $...
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2answers
75 views

When is the automorphism group of a nilpotent group nilpotent?

Let G be a group. I know that if $Aut(G)$ be nilpotent then $G$ is nilpotent also. Since $\frac{G}{Z(G)}‎\cong‎ Inn(G)‎\unlhd‎ Aut(G)$ Since $Aut(G)$ is nilpotent then $\frac{G}{Z(G)}$ is nilpotent....
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1answer
94 views

Non-nilpotent Engel group

A group $G$ is said to be $n$ engel if $[x,[x,\cdots,[x,y]]\cdots ]=1$ where $x$ appears $n$ times, and this holds for all $x,y\in G$. In 1955, P M. Cohn gave an example of non-nilpotent group which ...
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1answer
25 views

Conjugate of a nilpotent subgroup

Suppose that $G$ is a group with $H$ being a nilpotent subgroup of $G$. Let $g\in G$. Is is true that $gHg^{-1}$ is nilpotent in $G$?
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3answers
109 views

Nilpotent vs Solvability

I know that if G is solvable, then all subgroups and factor groups of G are solvable. I also know if N is normal in G, and N and G/N are solvable, then G is solvable.(which is kind of like the ...
2
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4answers
66 views

Is the class of all nilpotent groups closed under homomorphic images? [closed]

Suppose $H$ is a normal subgroup of a nilpotent group $G$. Does it imply, that $\frac{G}{H}$ is nilpotent? I do not know how to prove this. Could you help me, please?
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1answer
48 views

The periodic part of a locally nilpotent group.

Let $G$ be a locally nilpotent group and let $T$ be the periodic subgroup of $G$ (i.e., the locally nilpotent groups have a torsion subgroup). Then $T= \prod T_p$, where $T_p=\langle \{x \in T | \...
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0answers
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about minimal non-nilpotent groups

Newman and Wiegold have studied the AN-groups i.e. the locally nilpotent groups which are not nilpotent but every proper subgroup is nilpotent. I was asking why the notion locally nilpotent was added ...
3
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0answers
155 views

Nilpotent residual subgroup

Let $G$ be a finite group and $\gamma_{\infty}(G)$ be the limit of the lower central series: $\gamma_1(G)=G$ and for all $i\ge 1, \gamma_{i+1}(G)=[\gamma_i(G),G]$ ? It can be shown that $\gamma_{\...
3
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0answers
92 views

Characterizations of Nilpotent Groups

There are several characterizations of finite nilpotent groups (they are, as in wiki): $G$ is (finite) nilpotent group. Normalizer of every proper subgroup is bigger than the subgroup. Every ...
3
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3answers
847 views

Quotient of nilpotent group is nilpotent

Edit: I managed to rephrase my proof in a way that does not resort to coset multiplication. I think the resulting proof is better. I've added it as an answer below, while preserving the original ...
2
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1answer
201 views

Torsion-free nilpotent Group

I am looking for a short proof to the following fact: In torsion-free nilpotent group we have: an non-trivial element cannot be conjgate to it inverse. I know very little about nilpotent ...
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0answers
119 views

If $G=\langle\{a_i\}_{i=1}^r\rangle$ is nilpotent with $|a_i|=m_i$, show that $|G|$ divides a power of $\prod_{i=1}^rm_i$ and is finite.

use this notation for the following $\textbf{Theorem}$ - $\textbf{notation}-$ $G^n$ is the subgroup generated by $n$th power of elements of $G$ $\textbf{Theorem}$- In a finitely generated ...
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1answer
113 views

question on nilpotent group.

Question- If $G$ is a finite group, $N$ a normal nilpotent subgroup of $G$ such that $G/[N,N]$ is nilpotent. Prove that $G$ is nilpotent. How i did it in my exam today- (I know my solution had to be ...
5
votes
2answers
315 views

Does every two-generated subgroups being nilpotent imply that the group itself is nilpotent?

It is trivial that a group $G$ is abelian if and only if every subgroup of $G$ with two generators is abelian (i.e., any two elements commute). If $G$ is a nilpotent group, every subgroup with two ...
2
votes
1answer
79 views

Join of finitely many nilpotent subgroups (+ additional properties) is nilpotent?

I have the following: A group $X$ and a family $(X_n)_{n \in \mathbb{Z}}$ of subgroups of $X$, such that the following holds: $X = \langle X_n \;\vert\; n \in \mathbb{Z} \rangle$ $X_n \times X_{n+1} ...
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1answer
597 views

Commutator subgroup of a finitely generated nilpotent group.

Let $G$ be a finitely generated nilpotent group. Is the commutator subgroup $[G,G]$ finitely presented? Edit: I am also interested in the weaker question: is $[G,G]$ finitely generated?
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2answers
197 views

Does center contains commutator subgroup imply group is nilpotent?

Assume that $G^{'}\leq Z(G)$. Show that G is nilpotent. I show that if $G^{'}\leq Z(G)$ then $G/Z(G)$ is abelian and in particular $G/G^{'}$ is the largest abelian quotient group of $G$. but what I ...
4
votes
1answer
167 views

On nilpotent factor group

Let $G$ be a finite group and let $N$ be a normal subgroup of $G$ with the property that $G/N$ is nilpotent. Prove that there exists a nilpotent subgroup $H$ of $G$ satisfying $G = HN$. This is ...