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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I am stuck with this one problem related to nilpotent. Please guide me to write the proof.
I know how to go about when it is given that $L/Z(L)$ is nilpotent and we need to prove that $L$ is nilpotent. But how should I proceed if it is given that $L$ is nilpotent and we need to prove that $...
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If $G/Z(G)$ is nilpotent, then $G$ is nilpotent
If $G/Z(G)$ is nilpotent, then $G$ is nilpotent
Theorem 8 (Dummit, Foote) A group $G$ is nilpotent if and only if $G^n = \{1\}$ for some $n \geq 0$.
There is Quizlet solution to this problem which I ...
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A different approach to proving a property of nilpotent injectors in solvable groups
Let $G$ be a finite solvable group. Call $J\subseteq G$ a nilpotent injector if it is a nilpotent subgroup that contains $\mathbf{F}(G)$, and that is maximal with this property (not properly contained ...
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Is upper central series of a group unique?
Let G be a group. Its upper central series is defined inductively
$$Z_0=\{1\}$$
$$Z_{i+1}/Z_i=Z(G/Z_i)$$
Existence of $Z_{i+1}$ is guaranteed by correspondence theorem. But can we say anything about ...
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If $G$ is a finite group in which every subgroup has a normal complement, then $G$ is nilpotent.
Problem from an old prelim exam:
If $G$ is a group and $H$ is a subgroup, then a normal complement to
$H$ in $G$ is a normal subgroup $N\trianglelefteq G$ such that
$N\cap H = \{1\}$ and $N H = G$. ...
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References on Nilpotent groups of class 2
I wanted to read about the nilpotent groups of class 2, nilpotent groups of class 2 with exponent p and nilpotent groups of class q. Can anyone suggest some references (E.g., books, thesis, or ...
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Property preserved by tensor product/ epimorphic image/ extension [closed]
Let $G$ be a group. Consider $\gamma_1(G):=G$ and $\gamma_{i+1}(G):=[\gamma_i,G]$, we denote $G_{\text{ab}}$ the abelianization of $G$. Then $\gamma_{i+1}/\gamma_{i+2}(G)$ is an epimorphic image of $...
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Some calculation in $F_5[D_{30}]$.
I am currently reading a research paper and have encountered a point that I am struggling to understand. In the paper, it is proven that $J(F_5[D_{30}])^5 = (0)$, indicating that the Jacobson radical ...
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Commutator of two elements in group algebra $\mathbb F_{5}D_{30}.$
I want to understand how to find the commutator of two elements in the group algebra $\mathbb{F}_{5}D_{30}$ using GAP. Additionally, I would like to determine the nilpotency class of the nilpotent ...
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Non-Abelian group of exponent $3$ and Nilpotent class $2$ .
Let $G$ be a non-Abelian group such that $G^3 = 1$ and $G$ is a nilpotent class $2$ group, with order $3^{32}$. Our task is to determine the structure of the group $G$ or identify any information ...
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If $G$ is nilpotent of class $n$, then $G/Z(G)$ is nilpotent of class $n−1$.
I saw this here: If $G$ is nilpotent of class $c$, then $G/Z(G)$ is nilpotent of class $c-1$.
However, the definition of Nilpotent I have to work with is equivalent but different.
A group is Nilpotent ...
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Classification of torsion-free nilpotent groups of class 2
Some background
Let $G$ be a torsion-free nilpotent group of class $2$ and rank $2$ (i.e., generated by two elements). Then, $G$ has to be isomorphic to the Heisenberg group.
This is relatively easy ...
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What is the index of $H$ in $G=\langle H,g\rangle$, if $G$ is nilpotent and $g$ has finite order?
Suppose we have a nilpotent group $G$ that is generated by a subgroup $H$ and a single element $g$ of finite order. Is $H$ necessarily of finite index in $G$? What can be said of this index in any ...
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Class of nilpotency of semidirect product
Let $G=N\rtimes H$ for some groups $N$ and $H$. Is it true that the nilpotency class of $G$ is at most maximum of nilpotency classes of $N$ and $H$, if it is known that both groups are nilpotent? And ...
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Example of a quasi nilpotent element which is not a nilpotent element
Let $R$ be a ring with unity. An element $a\in R$ is said to be a quasi nilpotent element of $R$ if $1-ax$ is unit for all $x\in comm(a)$ where $comm(a)=\lbrace x \in R | ax=xa\rbrace $. It is obvious ...
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Nilpotent quotients of a residually nilpotent group
I am working on a problem that requires taking quotients by normal subgroups that do not intersect a finite set of members of a group $G$. Is there a known collection of groups which always allows ...
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Fitting length of Dihedral groups
Let $G$ be a dihedral group of order $2n$ where $n\geq 1$, denoted by $D_n$. We know that $G$ is nilpotent if and only if $n=2^i$ for all $i\geq 1$, a proof of this you can check in the below link 1. ...
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The $i^{th}$ term in the upper central series of the dihedral group of order $2^n$ is equal to the $(n-1-i)^{th}$ term in its lower central series.
The Problem: Prove that $Z_i(D_{2^n})=D_{2^n}^{n-1-i}$, where $D_{2^n}$ is the dihedral group of order $2^n$.
$Z_i(G)$ is the $i^{th}$ term in the upper central series of $G$, which is inductively ...
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Proof that if $G/Z(G)$ is nilpotent then $G$ is nilpotent.
As a reference my definition of nilpotence is as follows.
Definition: A group $G$ is said to be nilpotent if it has a subnormal series
$$
G = G_1 \trianglerighteq G_2 \trianglerighteq \dots \...
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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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Lie algebras of upper central series of connected Lie group
Let
\begin{equation*}
1=Z_0(G)\lhd Z_1(G)\lhd ...\lhd Z_{n-1}(G)\lhd Z_n(G)=G
\end{equation*}
be the upper central series of a connected nilpotent Lie group $G$, i.e. $Z_i(G)=\{g\in G:[g,G]\subset Z_{...
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An alternative notion of nilpotency class for $p$-groups
Let $G$ be a finite $p$-group for some prime $p$, and let $\rho: G \to \text{Aut}(A)$ a faithful representation of $G$ for some finite abelian $p$-group $A$ (which exists because, for example, we may ...
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Can we say anything about $\gamma_n(G) \cap \zeta_{n-1}(G)$, where $\gamma_n(G)$ is lower central series and $\zeta_{n-1}(G)$ is upper central series? [closed]
Can we say anything about $\gamma_n(G) \cap \zeta_{n-1}(G)$, where $\gamma_n(G)$ is lower central series and $\zeta_{n-1}(G)$ is upper central series?
For example one can say $Z(G)\cap [G,G]$ is a ...
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Reference request-dimension of projective representations of nilpotent groups.
There is folklore theorem concerning dimensions of complex projective representations of nilpotent groups that I want a reference for. I have searched Karpilovsky monographs but no success. The ...
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Let $G$ be a finite group. Suppose that there exists $n\in\mathbb N_0$ such that $G_n=G_{n+1}$ and $Z(G)=1$. Show that $C(G_n)\subseteq G_n$.
Let $G$ be a finite group. Recursively define the sequence $G_1=G$ and $G_{n+1}=[G_n,G]$. Suppose that there exists $n\in\mathbb N_0$ such that $G_n=G_{n+1}$ and $Z(G)=1$. Show that $C(G_n)\subseteq ...
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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$$ for every prime $p$.
Here,...
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Characterization of finite nilpotent groups
Can someone please check my proof of the following "evident" equivalence stated in Kurzweil and Stellmacher.
A finite group $G$ is nilpotent if and only if $$U<N(U) \text{ whenever } U&...
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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 ...
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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^...
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an Equivalent condition for nilpotency of a finite group like of finite solvable group
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 ...
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Is it true that the center of a finitely generated nilpotent group is finitely generated? [closed]
Let $G$ be a finitely generated nilpotent group.
Let $Z = Z(G)$ be the center of the group $G$.
Is $Z$ finitely generated?
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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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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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(renamed) Torsion free group has finite commutator subgroup 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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