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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About non-normal nilpotent subgroups

Suppose that all non-normal abelian subgroups of a finite group $G$ are cyclic. What can I say about non-normal nilpotent subgroups of $G$? Is it true that such supgroups are cyclic? I appreciate ...
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A $p$-group of exponent $p$

I saw a theorem For odd $p$, a $p$-group possesses a characteristic subgroup $D$ of class at most $2$ and of exponent $p$ such that every nontrivial $p’$-automorphism of $G$ induces a nontrivial ...
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can you help me about my algebra lesson [duplicate]

let n be a square free positive integer. prove that Zn has no nonzero nilpotent elements.
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Schur Zassenhaus theorem for nilpotent groups

I know that Schur Zassenhaus theorem is valid for any finite group, but my professor said that if a group is nilpotent group, the proof of Schur Zassenhaus theorem becomes very easy. However, I couldn'...
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Series of nilpotent group and one of its subgroups

Let $G$ be a nilpotent group of class $\leq n$ and $H\leq G$. There exists a sequence of subgroups $$G=H^1\supset H^2\supset\ldots\supset H^{n+1}=H$$ such that $H^{k+1}\trianglelefteq H^k$ and $H^k/...
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A finite product of nilpotent groups is nilpotent [duplicate]

Definition 1: Let $G$ be a group. The lower central series of $G$ is the sequence $\big(C^n(G)\big)_{n\ge 1}$ of subgroups of $G$ defined inductively by: $C^1(G)=G$ and $$C^{n+1}(G)=[G,C^n(G)],$...
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Centralisers in nilpotent Lie algebras

Let $K$ be a field of characteristic zero, let $\mathfrak{g}$ be a nilpotent Lie algebra over $K$, and let $\mathfrak{h}$ be a self-centralising abelian ideal of $\mathfrak{g}$, and we assume that $\...
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order of trace character distribution of irreducible unitary representations of nilpotent groups.

Let $N$ be a simply connected nilpotent group and let $\pi$ be an irreducible unitary representation of $N$. Then for every Schwartzfunction $f \in S(N)$ the integrated Operator $\pi(f)$ is of trace ...
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nilpotent uniform pro-$p$ groups of dimension 2

I read in a paper that all nilpotent uniform pro-$p$ groups of dimension $\leq 2$ are Abelian (Prime Decomposition and the Iwasawa $\mu$-Invariant by Hajir-Maire, Math. Proc. of the Camb. Phil. Soc. (...
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Do finitely generated nilpotent groups contain torsion free subgroups of finite index?

I have a question about the proof of proposition 6.9 of the paper "Rational Subgroups of Biautomatic Groups" by Gersten and Short (available here). The proposition states that a finitely presented ...
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Sylow Subgroups of a $2$-generated finite group

Let $G$ be a $r$-generated nilpotent finite group. I can write $G$ as: $G=\prod_{i=1}^r P_i \times \prod_{i=j}^l P_j$, where $p_1,...,p_r$ are exactly the prime divisors of $G$ for which the Sylow $p$-...
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Ascending central series for dihedral group $D_4$ of order 8

Assume the dihedral group of order $2n$ is defined by $$D_n=⟨s,r∣r^n=s^2=e,srs^{-1}=r^{-1}⟩$$ I know that $D_n$ is nilpotent iff $n=2^m$ for some $m\ge0$. I am asked to directly show that $G=D_4$ is ...
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Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$?

Does there exist some sort of classification of subvarieties of $\mathfrak{N}_2$? Here $\mathfrak{N}_2$ stands for the variety of nilpotent groups of class $2$ (defined by the identity $[[x, y], z]$). ...
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If a group has a normal subgroup of order $d$ for every divisor $d$ of the order of the group then it is nilpotent

How do I prove this? Let $G$ be a finite group. For every $d$ dividing $|G|$, there is a normal subgroup of order $d$. Then $G$ is nilpotent. The proof in my notes says that clearly every Sylow ...
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Dense subspace of $\mathcal{S}(G)$ for Nilpotent $G$

Let $G$ be a connected and simply connected nilpotent Lie group and let $\mathcal{S}(G)$ be it space of Schwartz functions, i.e. the image under $\exp$ of the Schwartz functions on its Lie algebra. ...
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Why isn't $Z_2 \times S_3$ nilpotent?

I have just learned the definition of a nilpotent group. My book seems to claim that $Z_2 \times S_3$ is not nilpotent, because they say, for the upper central series, $Z(G) = Z_1(G) = Z_2(G) = Z_n(G)$...
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A group of order 2295 is nilpotent.

Good evening, first of all sorry for my bad english and for my bad knowledge of algebra; so I need help to solve this exercise please. I want to show my idea (probably wrong): like in another Answer ...
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Fitting subgroups of infinite groups

Fitting subgroup is defined as the subgroup generated by all normal and nilpotent subgroups of a group G. If G is a finite group, we have that Fitting subgroup is nilpotent. If G is infinite not ...
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central series for a finite $p$-group

Define $G_2 = [G, G]$ and $G_{i+1} = [G_i, G]$ for each $i \geq 2$. Define $Z_1 =Z(G)$ and $Z_{i+1}$ is the unique subgroup of $G$ such that $Z_{i+1}/Z_i = Z(G/Z_i)$ for each $i \geq 1$. Let $p$ be a ...
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If $G$ nilpotent and $G/G'$ is cyclic then $G'=1$.

Hi: This question has already been answered here: Nilpotent group such that $G$/$G'$ is cyclic or a Prüfer group But I do not understand the proof given in the answer. $G$ is a ...
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Nilpotent Group G

Let $G$ be a finite group. $(1)$ If $G$ is nilpotent, then $\forall k \in \mathbb{Z} : k \mid |G|$, $\exists H \lhd G : |H| = k$. $(2)$ If $\forall k \in \mathbb{Z} : k \mid |G|$, $\exists!...
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Equivalent condition for finite Nilpotent group [duplicate]

Prove that a finite group $G$ is nilpotent if and only if whenever $a, b \in G$ with $gcd(o(a), o(b)) = 1$, then $ab = ba$. I know that a finite group G is nilpotent iff it is a direct product of its ...
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A commutator with an element of finite order in a nilpotent group

Let $G$ be a group. We let $G_1:=G$, we let $G_2 := [G,G]$ the commutator subgroup (i.e. the subgroup generated by all the $[a,b]$ such that $a,b\in G$). Inductively one can define $G_k = [G_{k-1},G]$....
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Proving a necessary and sufficient condition for a finite group being nilpotent

I was solving the following question : A finite group $G$ is nilpotent if every proper maximal subgroup of $G$ is normal. [Hint: If $P$ is a Sylow $p$-subgroup of $G$, show that any subgroup ...
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Conjugacy classes in virtually nilpotent groups

Let $G$ be a f.g. virtually nilpotent group. Can an element $g\in G$ of infinite order be conjugate to its power $g^n$ for $n>1$? Let $G$ be a f.g. virtually abelian group. Is it true that ...
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Free nilpotent group on 2 generators of class 2.

Classical presentation of Heisenberg group is [x,y,z|[x,y]=z, xz=zx, yz=zy] https://pdfs.semanticscholar.org/276f/aef6c6b534f6058441a4f96d6260c5f32052.pdf Here above on page 12 it is written that we ...
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Commutator subgroup of Heisenberg group.

Dears, Let $H$ be Heisenberg group, a group of $3\times 3$ matrices with $1$ on the main diagonal, zeros below, and elements of $\Bbb R$ above the main diagonal. Its center is the subgroup of all ...
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Are all nilpotent groups hamiltonian?

Are all nilpotent groups hamiltonian? That is, is every subgroup of a nilpotent group normal? I don't think so. Every Sylow subgroup of nilpotent group is normal and every nilpotent group is a direct ...
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Commutator power lying in commutator subgroup

The argument used in the proof of Proposition 3 of this math.SE answer appears to prove the following claim: Let $G$ be a group and let $H\subseteq G$ be a normal subgroup. Let $n\geq 0$ and let $x,...
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Proving that a quotient is virtually a nilpotent group

Let $G$ be a group and let $L$ be a normal subgroup of $G$. Moreover, I have normal finite index subgroups $N_{i}$ of $G$ for $i\in \{1,\cdots,n\}$ such that the $n$-fold commutator $$[N_{1},\cdots,N_{...
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Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite.

Given: Let $G$ be a nilpotent group generated by a finite set of torsion (i.e. finite order) elements. Show that $G$ is finite. Also would love to know if it's possible to show that an infinite ...
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Question regarding writing a group of order $p^2qr$ using notations

Let $G$ be a solvable, non-nilpotent group of order $p^2qr$, where $p,q,r$ are distinct primes, and let $F$ be a Fitting subgroup of $G$. Then $F$ and $G/F$ are both non-trivial and $G/F$ acts ...
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Are Triangle Groups virtually nilpotent?

I'd like to know the Growth Function of the Triangle Groups. From the Gromov's theorem we know that every virtually nilpotent group has a polynomial growth. It seems that T(2,4,4) is virtually ...
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Reduce completion time of projects with constraints

I have a Operation Research Problem in which the aim is to reduce the completion time of projects. For explanation: consider I have projects as A, B, P1, P2, C. Such that projects A, B are inputs to ...
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A polynomial algorithm to determine whether a finite group is nilpotent

Does there exist a polynomial (in respect to the order of the group) algorithm that given a Cayley table of a finite group determines, whether a group is nilpotent or not? There do exist polynomial ...
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Basic theorem for solvable groups not true for nilpotent groups - counterexample.

it's my first question on MathStackExchange so please be tolerant. Let H be a normal subgroup of group G. If H and G/H are both solvable, then G is solvable. But H nilpotent and G/H nilpotent doesn't ...
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On the nilpotency class of a certain subgroup [closed]

Let $G$ be a nilpotent group of nilpotency class $c>2$ and $a\in G‎\setminus G'$. Is the nilpotency class of $\langle a\rangle G'$ less than c?
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Nilpotency class of Frattini subgroup and group order

Suppose $\psi(n)$ denotes the minimal natural number $k$, such that there exists a finite group $G$, such that $k = \max \{m \in \mathbb{N}| \exists \text{ prime } p, p^m | |G| \}$, and $\Phi(G)$ has ...
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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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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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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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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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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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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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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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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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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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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 ...