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 ...
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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 ...
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Each element of a nilpotent group is nilpotent

Suppose $G$ is a nilpotent group, i.e. it has an upper central series: \begin{align} Z_{0}(G) \leq Z_{1}(G) \leq Z_{2}(G) \leq ... \end{align} where $Z_{i+1}(G) = \{x \in G : [x,y] \in Z_i(G) \, \...
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On nilpotent groups of class 2

So I have a simple question: I would like to know if the converse of this question here is also true, that is, if $G$ is a group with $G'\leq Z(G)$ then $G$ is a nilpotent group of class 2. I'm ...
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What are some "easy, transcendental" constructions of finite nilpotent groups?

I considered this question in teaching abstract algebra. I am trying to impress upon my students about the messiness of studying nilpotent groups generally. More specifically, I am looking for ways ...
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Commutator subgroup of a Wang group is finite iff abelian

We say a group $ G $ is a Wang group if it fits into a SES $$ 0 \to N \to G \to \mathbb{Z}^k \to 0 $$ where $ N $ is nilpotent finitely generated and torsion free. Such a group $ G $ is always ...
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Self-centralizing subgroups of order $p^2$ in $p$-groups of maximal class

Let $G$ be a finite $p$-group. It's an old result of Suzuki that if $G$ possesses a self-centralizing subgroup of order $p^2$ then $G$ has maximal class, and in fact the converse is true. However, the ...
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If $N\unlhd G$ is nilpotent and $G/N'$ is supersolvable, then $G$ is supersolvable.

This is Exercise 5.4.7 of Robinson's "A Course in the Theory of Groups (Second Edition)". According to this search, it is new to MSE. The Details: (This can be skipped.) Denote the derived ...
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Proof Verification: $G/Z(G)$ is nilpotent implies $G$ is nilpotent

I wrote a proof down for the fact that if $G/Z (G)$ is nilpotent then $G$ is nilpotent and I wonder if this is correct. I would appreciate it if someone verifies this and/or suggests some alternative ...
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If $N$ is normal in $G$, show $Z_{i}(G)N/N \leq Z_{i}(G/N)$ where $Z_{i}(G)$ is the $i$th term in the upper central series for $G$.

If $N$ is normal in $G$, show $Z_{i}(G)N/N \leq Z_{i}(G/N)$ where $Z_{i}(G)$ is the $i$th term in the upper central series for $G$. The argument is supposed to be by induction, and I have taken care ...
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If $N$ is a normal subgroup of $G$, show $Z(G)N/N \subset Z(G/N)$ [closed]

My notes claim that $Z(G)N/N = \{ zN \in G/N :[g,z] \in N \;\forall g \in G \}. ​$ It is clear to me that the right hand side is contained in $Z(G/N)$, however I cannot make sense of the equality to ...
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Characterization of nilpotency with normal subgroups

I have to prove the following: Let G be a finite group then G is nilpotent $\iff$ every proper normal subgroup, $N\lhd G$, satisfy $[N,G]\leq N$ But it's not true that for every normal subgroup (even ...
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Show $G=\langle\delta\rangle\ltimes D$ is nilpotent of class $2$.

This is part of Exercise 5.2.2(a) of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. It is marked as being referred to later on in ...
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If a nilpotent group has an element of prime order $p$, so does its centre.

This is Exercise 5.2.1 of Robinson's, "A Course in the Theory of Groups (Second Edition)". According to Approach0, it is new to MSE. The closest I could find is the following: Prove that in ...
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If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? [duplicate]

If $G$ nilpotent and $G/G'$ is cyclic then $G$ is cyclic? It is very easy to see that this is true when $G$ is finite: If $G$ is finite nilpotent then all maximal subgroups are normal, so $G/N$ has to ...
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If all proper subgroups of $G$ are nilpotent and $G$ is solvable, the $G$ is nilpotent.

I will assume that $G$ is finite. If all proper subgroups of $G$ are nilpotent, believe that it can be proven quite easily that $G$ is soluble. But since I don't want to write the details of the proof ...
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why "this means that $G'' \subset Z(G')$"?

I am trying to understand the proof of the following question: Show that if $G'/G^{''}$ and $G^{''}/G^{'''}$ are both cyclic then $G^{''} = G^{'''}.$[you may assume $G^{'''} = 1.$ Then $G/G^{''}$ acts ...
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Lower central series of the Unitriangular group $UT(n, \mathbb{Z}_p)$

This is Exercise 5.44 from Rotman's book "An Introduction to the theory of Groups (4th Ed)". Specificaly, the exercise asks us to prove that the $i$-eth term in the lower central series is ...
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Suppose $T(B)=AB+BA$. Prove that if $A$ is a nilpotent matrix, then $T$ is a nilpotent operator. (Question on Existing Proof)

Question: Suppose $A$ is a complex $n\times n$ matrix and let $T:\mathbb{C}^{n\times n}\rightarrow\mathbb{C}^{n\times n}$ be the linear transofrmation given by $T(B)=AB+BA$ for $B\in\mathbb{C}^{n\...
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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 ...
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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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2 votes
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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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Is every group of order $136$ nilpotent?

Is every group of order $136$ nilpotent? The answer is no. Consider the group $G=\mathbb{Z}_{17}\rtimes_\phi \mathbb{Z}_8$ where $\phi: \mathbb{Z}_8 \to {\rm Aut}(\mathbb{Z}_{17})\simeq \mathbb{Z}_{...
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p-nilpotent group and p-Fitting subgroup

Let $G$ a finite group. I use notation on this reference for $O_p(G), O_{p'}(G), O_{p',p}(G)$. Furthermore I define the $p$-residual $O^p(G)$ as the minimum normal subgroup with quotient $\frac{G}{O^p(...
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An association between finite nilpotent groups and Lie rings

The following line turns up in a paper without much of a reference: Let $G$ be a nilpotent groups of nilpotency class $c$, and suppose $V$ be its associated Lie ring, where $V=V_1\oplus V_2\cdots \...
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Direct Product of a nilpotent group with a supersolvable group is supersolvable

Let $H$ be a nilpotent group and $K$ a supersolvable group. Considering $G=H \times K$, then $G$ is supersolvable. I tried to create a normal cyclic series, but I don't know how to proceed, maybe the ...
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Given $p$ prime and $H\le S_p$ s.t. $|H| = p$, show if $H\subseteq N\subseteq S_p$ for some nilpotent subgroup $N$, then $H=N$

Problem Statement: 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 $H=N$. By working with conjugacy classes and orbit-...
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6 votes
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If $G$ is a nilpotent group and $H\leq G$ with $H[G,G]=G$ then $H=G$.

I am studying for a qualifying exam and this problem has been a white whale. Let $G$ be a nilpotent group with subgroup $H\leq G$. If $H[G,G]=G$, then $H=G$. I believe I should use the fact that if $...
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Proving nilpotency class of certain subgroup is less than nilpotency class of group

Suppose $G$ is a nonabelian nilpotent group and let $x \in G$. I am trying to show that $\langle [G,G], x \rangle$ is a proper subgroup of $G$. If I can show that the nilpotency class of $\langle [G,G]...
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Let $G$ be a finite group, $H\leq G$ s. t. $\forall g\in G$, either $\langle H, H^g \rangle$ is nilpotent or $H H^g = H^g H$. Show that $H \lhd\lhd G$

Let $G$ be a finite group, $H\leq G$ such that $\forall g\in G$, either $\langle H, H^g \rangle$ is nilpotent or $H H^g = H^g H$. Show that $H \lhd \lhd G$. I know how to show that if $\langle H,H^g \...
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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 ...
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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 ...
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2 votes
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Nilpotent quotient of semidirect product of a nilpotent group and a free abelian group

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 ...
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Let $H=\langle\sigma\rangle$ be the infinite cyclic group. Consider $G = (\mathbb{Z}\times \mathbb{Z})\rtimes H$. Determine $G'$.

Let $H=\langle\sigma\rangle$ be the infinite cyclic group. Consider $G = (\mathbb{Z}\times \mathbb{Z})\rtimes H$ with action given by $$ \sigma:\mathbb{Z}\times\mathbb{Z}\to \mathbb{Z}\times\mathbb{Z}:...
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If $G$ is a nilpotent torsion group of nilpotence class $c$, then $\exp(G)$ divides $\exp(G^{\operatorname{ab}})^c$

I am trying to prove the following statement: If $G$ is a torsion group and nilpotent of class $c$, then $\exp(G)$ divides $\exp(G^{\operatorname{ab}})^c$ (where $\exp$ is the exponent, i.e. the lcm ...
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2 votes
3 answers
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Why Nilpotent Matrix is not Null matrix always?

I know it might sound dumb, but specifically, Why NULL matrices are not the only NULPOTENT matrices? I am thinking that as all eigen values of NILPOTENT matrices are 0, then $\lambda = 0$, and as per ...
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6 votes
3 answers
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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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4 votes
2 answers
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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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2 votes
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Recursive expression for $\Lambda(n)$: number of nilpotent groups of order $n < \infty$

My question is related to this one here about nilpotent groups, but it is different since I want to prove an explicit expression for the number of nilpotent groups. Suppose $n = p_1^{k_1}p_2^{k_2}\...
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2 votes
1 answer
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About nilpotent groups of orders $1$ to $59$ up to isomorphism

Find the number of all nilpotent groups of order $<60$, up to isomorphism - i.e. for every $n \in \{1,2,\ldots,59\}$, find the number of nilpotent groups up to isomorphism. We know that Result 1: ...
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3 votes
1 answer
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Intuition behind Definition of Nilpotent Groups

Definition: A group $G$ is called nilpotent if there exists a chain of subgroups $N_0, N_1,\ldots, N_k$ such that $$\{e\} = N_0 \le N_1 \le N_2 \le ... \le N_k = G$$ and for $0\le i\le k-1$, $N_i \...
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1 vote
2 answers
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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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