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].

Filter by
Sorted by
Tagged with
5
votes
3answers
116 views

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 ...
4
votes
2answers
153 views

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 ...
2
votes
1answer
47 views

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 ...
1
vote
1answer
66 views

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}\...
2
votes
1answer
173 views

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: ...
3
votes
1answer
70 views

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 \...
1
vote
2answers
72 views

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 ...
1
vote
0answers
19 views

Is every simply-connected nilpotent Lie group algebraic?

Say a Lie group is a matrix group if it is a closed Lie subgroup of some general linear group. Say a Lie group is algebraic if it is the group of $\mathbb R$-points of a real linear algebraic group. ...
3
votes
2answers
69 views

Number of conjugacy classes of a group of order $5^4$ whose center is $25$ order

How many conjugacy classes does a group of order $625$ have, if its center is of order $25$? I know that the orders of conjugacy classes should divide the order of the group, so this leaves with each ...
1
vote
0answers
50 views

How to show that $G$ is nilpotent based on these conditions

Am trying to figure out a proof that if $G$ is a finite $p$-group, then $G$ is nilpotent ... Here, nilpotent means that a group $G$ is nilpotent if there exists a series $1 = H_0 \leq H_1 \leq ... \...
0
votes
1answer
49 views

If $G$ is a finite group such that every maximal subgroup is nilpotent, are any two maximal subgroups of $G$ conjugate?

I’m trying to prove that if every maximal subgroup of a finite group $G$ is nilpotent, then $G$ is solvable. I know already that if any two maximal subgroups of a finite group are conjugate, then it ...
2
votes
0answers
53 views

$G$ is nilpotent iff $[N,G]<N$ for every nontrivial normal subgroup of $G$.

I need help to prove the following statement: “A finite group $G$ is nilpotent iff $[N,G]<N$ for every nontrivial normal subgroup of $G$.” I’ve found a proof for the forward direction on the ...
1
vote
2answers
90 views

Relationship between the center and commutator subgroup of a group of nilpotency class 3

I am new to the theory of nilpotent groups. I am dealing with the nilpotent groups of class 3. I want to know is there any relationship between the commutator subgroup $[G, G]$ of the group $G$ and ...
1
vote
1answer
24 views

Almost nilpotent groups of hyperbolic isometries are cyclic

Let $X$ be a complete, simply connected, Riemannian manifold of negative curvature. Let $\Gamma$ be an almost nilpotent group of non-elliptic isometries containing a hyperbolic isometry $\gamma$. Then ...
3
votes
1answer
26 views

ith Center of Product is Product of ith Centers

The Lemmas: -Lem. 1: Let $k\in\mathbb{N}$. Let $G_{1}$, ..., $G_{k}$ be groups. Then $$Z(G_{1}\times...\times G_{k})=Z(G_{1})\times...\times Z(G_{k}).$$ -Lem. 2: Let $k\in\mathbb{N}$. For $1\leq j\leq ...
1
vote
2answers
63 views

Nilpotent groups can be constructed by means of abelian groups.

I am studying A course in theory of groups by Robinson. When defining the central extension of groups, the author says that Every nilpotent group can be constructed from abelian groups by means of a ...
0
votes
1answer
58 views

Having trouble understanding nilpotent and supersolvable group in terms of short exact sequence

$G$ is supersolvable if $G_{i+1}/G_i$ is cyclic for every $i\geq 0$ where $G_i$ is a normal sequence of $G=G_n$. According to the lecture note I read, it says that in the following short exact ...
1
vote
1answer
35 views

Semidirect product is supersolvable groups

Let $G$ be a finitely generated group and $N$ be a nilpotent group (not necessarily finitely generated) such that $N$ is a subgroup of $G$ and $G / N$ is $\mathbb{Z}$. That is, there is a short exact ...
5
votes
1answer
61 views

Characterization of a virtually nilpotent group

Let $G$ be a group that has an increasing sequence of subgroups $G_i \le G_{i+1}$ satisfying the following properties. (a) $G=\bigcup_{i\ge 1} G_i$. (b) Each $G_i$ contains a nilpotent subgroup $N_i$ ...
-1
votes
1answer
34 views

The rank of torsion free nilpotent group [closed]

Let $N$ be a torsion free finitely generated nilpotent group. We define the rank of $N$ to be the natural number $m$ such that there exists a finite subnormal series $$ N=N_m \vartriangleright N_{m-1} ...
1
vote
0answers
37 views

Give an example of a solvable group of prime exponent which is not nilpotent.

I came across two examples on Wikipedia -"McClain's Example" and "Tarski Group" ; but of them looked quite complicated .Are there any other somewhat simpler examples which satisfy ...
0
votes
1answer
55 views

Every (finite) group having a central series is nilpotent.

The statement is from Kurzweil and Stellmacher, The Theory of Finite Groups, An Introduction, as well as the following: (a) A group $G$ is nilpotent if every subgroup of $G$ is subnormal in $G$. ...
-1
votes
1answer
56 views

Nilpotent Matrices Questions.

Nilpotent Matrix Question Link I was wondering if anyone could help with the latter parts of the question (b & c). I have concluded from part A that Matrix "A" is Nilpotent as det(A)=0 ...
0
votes
0answers
78 views

Is this an equivalent definition of a nilpotent group?

I'll try to make it quick. In an exercise, I'm encountering this definition of a nilpotent group: A group $G$ is nilpotent if there is a finite series of subgroups $\{e\} \subseteq G_n \subseteq G_{n ...
-2
votes
2answers
79 views

Show that if an $n \times n$ matrix, $A$, is nilpotent, then $\text{rank}(A) < n$

A matrix is $A$ is nilpotent if $A^k$ is the $0$ matrix for some $k$. Show that if an $n \times n$ matrix $A$ is nilpotent, then $\text{rank}(A) < n$. Anyone got any ways to get started on this ...
0
votes
1answer
78 views

Let $G$ be a nilpotent group, $H\subsetneq G$. How to prove $H \subsetneq N$, where $N$ is a normalizer of $H$? [closed]

Let $G$ be a nilpotent group, $H\subsetneq G$. How to prove $H \subsetneq N$, where $N$ is a normalizer of $H$? I tried to use induction. It's easy to show the property works when nilpotency is equal ...
8
votes
0answers
76 views

A generalizaton of nilpotent groups

‎‎‎‎‎‎‎Let ‎$‎G‎$ ‎be a‎ ‎group ‎and ‎‎$‎‎\alpha\in Aut(G)$ ‎be a‎ ‎fixed ‎automorphism ‎of ‎‎$‎G‎$‎. An ‎$‎‎\alpha$-commutator ‎of ‎elements ‎‎$‎‎x, y\in G$ ‎is ‎‎$‎‎[x, y]_{\alpha}= x^{-1}y^{-1}xy^{\...
1
vote
1answer
29 views

An Infinite non-nilpotent group whose every maximal subgroup is a normal subgroup.

It is widely known that for every finite group $G$, $G$ is nilpotent If and only if every maximal subgroup of $G$ is a normal subgroup. But I don't know if there is an infinite non-nilpotent group ...
4
votes
1answer
119 views

Finite nilpotent groups

Let $G$ be a finite nilpotent nonabelian group. Is it true that for every natural number $k$ there exists a finite group $G_k$ such $G_k$ is not isomorphic to a subgroup of a direct power of $G$ while ...
7
votes
3answers
140 views

Let $G$ be a finite nilpotent group and $G'$ its commutator subgroup. Show that if $G/G'$ is cyclic then $G$ is cyclic.

So I thought the cleanest way to do this was to simply prove $G' = 1$ since if $G$ is cyclic $G' = 1$ and then $G \cong G/G'$, but I got no where with this. My next idea was since $G$ is nilpotent I ...
4
votes
2answers
351 views

What exactly does the definition of a nilpotent group mean?

I'm studying nilpotent and solvable group and find it pretty hard to tell what the definition of a nilpotent group is after. For example, a group is solvable iff it has a solvable series (that is, a ...
5
votes
1answer
495 views

Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable.

Prove that if $G$ is a finite group in which every proper subgroup is nilpotent, then $G$ is solvable. (Hint: Show that a minimal counterexample is simple. Let $M$ and $N$ be distinct maximal ...
1
vote
1answer
66 views

Proving Jordan chain for nilpotent matrix is linearly independent

I am looking for a proof for the Jordan chain $\{L^ix, L^{i-1}x, \dots, x\}$ being independent, where $L$ is a nilpotent matrix of index $k$, $i<k$ and $x \neq 0$. I have tried the following. ...
1
vote
0answers
38 views

Finitely Presented Group of order $p^4$

In the (xi) group of the classification of groups of order $p^4$ given by W.Burnside in his book," Theory of Groups Of Finite Order". The group ($\mathbb{Z_{p^{2}}}\rtimes \mathbb{Z_{p^{}}}) ...
2
votes
1answer
117 views

Heisenberg group is nilpotent

$F$ is a field and $H(F)$ is the Heisenberg group over $F$. Is it nilpotent? Is it solvable? I did all the math and I found that the commutator subgroup is in the center $Z(H(F))$, so $H(F)/Z(H(F))$ ...
1
vote
2answers
38 views

Index of commutator subgroup in the commutator group

I need to prove (or find a counter example) that if $G$ is solvable and $H \leq G$ is a subgroup of finite index, then the commutator subgroup $D(H)$ is also a subgroup of finite index in $D(G)$. This ...
4
votes
1answer
55 views

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 ...
0
votes
1answer
111 views

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 ...
1
vote
1answer
40 views

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'...
1
vote
1answer
31 views

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/...
2
votes
0answers
48 views

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)],$...
4
votes
2answers
72 views

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 $\...
0
votes
0answers
44 views

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 ...
1
vote
1answer
43 views

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. (...
5
votes
1answer
178 views

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 ...
3
votes
1answer
359 views

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 ...
5
votes
1answer
72 views

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 $[[x, y], z]$). ...
3
votes
2answers
159 views

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 ...
1
vote
1answer
83 views

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, $Z(G) = Z_1(G) = Z_2(G) = Z_n(G)$...
1
vote
0answers
44 views

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 ...