Questions tagged [nilpotent-groups]

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Is every $p$-group the $\mathbb{F}_p$-points of a unipotent group

Let $\Gamma$ be a finite group of order $p^n$. Is there necessarily a unipotent algebraic group $G$ of dimension $n$, defined over $\mathbb{F}_p$, with $\Gamma \cong G(\mathbb{F}_p)$? I have no real ...
David E Speyer's user avatar
24 votes
2 answers
1k views

Nilpotency of a group by looking at orders of elements

For any finite group $G$, let $$\theta(G) := \sum_{g \in G} \frac{o(g)}{\phi(o(g))},$$ where $o(g)$ denotes the order of the element $g$ in $G$, and where $\phi$ is the Euler totient function. It is ...
Tom De Medts's user avatar
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15 votes
1 answer
639 views

Linear embeddings of nilpotent pro-$p$ groups

Is it true that every finitely generated (topologically) torsion-free nilpotent pro-$p$ group is isomorphic to a subgroup of $U_d(\mathbb{Z}_p)$, the group of $d\times d$-upper triangular matrices ...
Diego Sulca's user avatar
14 votes
3 answers
1k views

Explicit formulas for Carnot-Carathéodory distances on Carnot groups

Let $G$ be a Carnot group (aka stratified group), so that $G$ is a connected and simply connected finite-dimensonal Lie group, whose Lie algebra $\mathfrak{g}$ admits a decomposition $\mathfrak{g} = ...
Nate Eldredge's user avatar
13 votes
1 answer
440 views

Variety of nilpotent Lie algebras or $p$-groups

Here's a couple of analogous questions, one in terms of finite-dimensional complex Lie algebras and one in terms of finite $p$-groups; I'd be interested in an answer to either: 1) Let $\mathcal{L}$ ...
YCor's user avatar
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13 votes
1 answer
728 views

Characteristic subgroup of nilpotent group that is not invariant under powering

I want an example of a nilpotent group $G$, a characteristic subgroup $H$, and a prime number $p$ such that: $G$ is $p$-powered, i.e., every element of $G$ has a unique $p^{th}$ root in $G$. $H$ is ...
Vipul Naik's user avatar
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12 votes
3 answers
2k views

Finite index subgroup with free abelianization

This question has been asked on MathExchange to no avail. Suppose $G$ is a finitely generated nilpotent group with abelianization of rank $r$. Does $G$ always have a subgroup $H$ of finite index, ...
Matthew Clayton's user avatar
12 votes
1 answer
2k views

Some questions about the Malcev completion

Let $G$ be an abstract group. The Malcev completion $\widehat{G}$ of $G$ (over $\mathbb{Q}$) is the set of group-like elements in the complete Hopf algebra $\widehat{\mathbb{Q}[G]} = \lim_n \mathbb{Q}...
Mostafa's user avatar
  • 4,454
11 votes
2 answers
721 views

Quasi-isometric rigidity of Nil

Let $Nil$ be the unique simply connected non-abelian three-dimensional nilpotent Lie group, i.e. the group of upper triangular matrices with all the eigenvalues equal to 1 (this group is also known as ...
Roberto Frigerio's user avatar
11 votes
0 answers
248 views

Can the orbit with respect to a rotation on the torus hit an algebraic variety infinitely often?

Question: Does there exist a $d$-tuple $\alpha = (\alpha_1,\dots,\alpha_d) \in \mathbb{R}^d$ (with $1,\alpha_1, \dots,\alpha_d$ linearly independent over $\mathbb{Q}$) and an algebraic variety $V \...
Jakub Konieczny's user avatar
10 votes
1 answer
547 views

Can automorphism equivalence in a free group be detected in a nilpotent quotient?

If $G$ is a group and $g_1, g_2 \in G$ let us write $g_1 \sim g_2$ if there is an automorphism $\alpha \in \operatorname{Aut}(G)$ such that $g_1^\alpha = g_2$. Let $F = F_2$ be the free group on two ...
Sean Eberhard's user avatar
9 votes
2 answers
229 views

Residual finiteness for modules over group rings

Let $G$ be a finitely generated residually finite group and let $M$ be a finitely generated $\mathbb{Z}[G]$-module. Question: Must $M$ be residually finite in the sense that for all nonzero $x \in M$, ...
Alice's user avatar
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9 votes
1 answer
606 views

What is the simplest known finite presentation of a free nilpotent group?

Finitely generated nilpotent groups are always finitely presented. This is true for abelian groups, and can be shown by induction for nilpotent ones, using the classical lift of a presentation of $N$ ...
J. Darné's user avatar
  • 263
9 votes
0 answers
428 views

(Torsion in) homology of free nilpotent groups

It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). ...
Denis T's user avatar
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8 votes
1 answer
834 views

Largest possible order of a nilpotent permutation group?

I'm trying to obtain a bound for the order of some finite groups, and part of it comes down to the order of a permutation group of degree $n$ that is nilpotent. I imagine these have to be much ...
Colin Reid's user avatar
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7 votes
5 answers
3k views

Nilpotent group with ascending and descending central series different?

This may turn out to be a bit embarassing, but here it is. It is probably not true that the ascending and descending central series (*) of a nilpotent group have the same terms. (Or at least one of ...
Pietro's user avatar
  • 1,570
7 votes
2 answers
847 views

Quotients of finitely generated nilpotent groups

Is the following fact true? Let $N$ be a finitely generated nilpotent group, and denote its lower central series by $(N_r)_{r\ge 1}$, that is, $N_1=N$ and $N_{k+1}=[N,N_k]$ is the ...
Miel Sharf's user avatar
7 votes
1 answer
269 views

Lower central series vs torsion-free lower central series

$\newcommand\tf{\text{tf}}\newcommand\tor{\text{tor}}$Let $G$ be a finitely generated group. Let $\gamma_k(G)$ denote the $k$th term in the lower central series for $G$, so $\gamma_1(G) = G$ and $\...
Irina's user avatar
  • 437
7 votes
1 answer
174 views

Universal enveloping algebra of Malcev Lie algebra associated to nilpotent group

Let $G$ be a finitely generated torsion-free nilpotent group. The Malcev completion of $G$ is a nilpotent Lie group $N$ into which $G$ embeds as a lattice. One way to construct this is to take the ...
Roberta's user avatar
  • 153
7 votes
1 answer
246 views

Subgroups of Nilpotent groups with prescribed center

Let $G$ be a torsion-free, finitely-generated, nilpotent group of nilpotency class at least 3. Does there exist a normal subgroup $N\leq G$ such that $G/N\cong \mathbb{Z}$ and $Z(G)=Z(N)$? (By $Z(H)...
Caleb Eckhardt's user avatar
7 votes
0 answers
181 views

Growth of spheres in FINITE nilpotent groups - Gaussian approximation (central limit theorem)?

Standard setup. Consider a group and choose generators. Word-metric (or in the other words - distance on the Cayley graph of the group+generators) - converts a group into a metric space, which is ...
Alexander Chervov's user avatar
7 votes
0 answers
420 views

Reconstructing a nilpotent Lie algebra from its cohomology with $A_{\infty}$-structure

Let $L$ be a nilpotent Lie algebra (over a field of char 0) and $CE^{\bullet}(L)$ be its Chevalley-Eilenberg dg-algebra. By homotopy transfer, there exists a structure of an $A_{\infty}$-algebra on ...
Grisha Papayanov's user avatar
7 votes
0 answers
1k views

Computational complexity of multiplication in a nilpotent group?

What is the computational complexity of multiplication in a Carnot group ? Background: A Carnot group is a real nilpotent Lie group $N$ whose Lie algebra $Lie(N)$ admits a direct sum decomposition ...
Marius Buliga's user avatar
6 votes
1 answer
226 views

Do lattices of small co-volume always exist in rational, connected, simply connected, nilpotent Lie groups?

Given a connected, simply connected, rational, nilpotent Lie group $G$, is there a lattice of arbitrarily small co-volume in $G$? If $G$ is Carnot, the answer is "yes" by applying a ...
Kyle's user avatar
  • 243
6 votes
1 answer
572 views

Why do we say the Fitting subgroup/generalized Fitting subgroup control the structure of a group?

I’m learning the Fitting subgroup these days. I’m interested in this topic and particularly in the role that it plays in the structure of groups. Many people on MSE mentioned that the Fitting subgroup/...
user avatar
6 votes
1 answer
234 views

Betti numbers and lower central series quotients of finite-index subgroups of nilpotent groups

Let $G$ be a finitely generated nilpotent group and let $A\le G$ be a finite-index subgroup. I have two questions about $A$: Is it true that the inclusion map $A \rightarrow G$ induces isomorphisms $...
Irina's user avatar
  • 437
6 votes
1 answer
230 views

When is a nilpotent Lie algebra isomorphic to the associated graded of its lower central series?

All Lie algebras in this question are finite-dimensional and defined over a field $k$ of characteristic $0$ which I'm happy to take to be $\mathbb{R}$ or $\mathbb{C}$. $\DeclareMathOperator\gr{gr}$Let ...
Irina's user avatar
  • 437
5 votes
2 answers
491 views

Maximal $p$-subgroups in nilpotent groups

By using Zorn's Lemma one can establish the existence of maximal $p$-subgroups in any group, even infinite. Using this existence, exactly as in the finite case, it is easy to show that in a nilpotent ...
Ali Nesin's user avatar
5 votes
1 answer
218 views

Finite solvable groups are generated by a nilpotent subgroup + K elements?

Is there a constant $K \in \mathbb{N}$ such that for every finite solvable group $G$, there exists a nilpotent subgroup $N \leq G$, and a subset $S \subseteq G$ with $|S| \leq K$, and $\langle N,S\...
Pablo's user avatar
  • 11.2k
5 votes
2 answers
927 views

Malcev's paper "On a class of homogeneous spaces" in English

I am struggling to find the English translation of Malcev's paper "On a class of homogenous spaces" providing foundational material for nil-manifolds. To be precise this paper: Malcev, A. I. On a ...
Tom1990's user avatar
  • 51
5 votes
1 answer
208 views

Local vs global nilpotence class (Lazard correspondence)

The Lazard Correspondence is often phrased (for simplicity) for $p$-groups of nilpotence class $c < p$, but it works more generally whenever every 3-generated subgroup has nilpotence class $< p$,...
Joshua Grochow's user avatar
5 votes
1 answer
171 views

Example of an invariant metric on a nilpotent group which is not asymptotically geodesic

Let $X$ be a metric space. We say that $X$ is asymptotically geodesic if for all $\epsilon > 0$, there exists $R > 0$ such that, for all $x,y \in X$, there exists some finite sequence of points $...
Christian Gorski's user avatar
5 votes
1 answer
250 views

Harmonic analysis on nilpotent Lie groups and the Campbell-Hausdorff formula

I am trying to understand the non-commutative analysis for nilpotent Lie groups, so I've been reading Corwin's and Greenleaf's book on the representation theory of nilpotent groups and going through ...
Ivan's user avatar
  • 445
5 votes
0 answers
345 views

Adjoint identity on finite nilpotent groups

Let $G$ be a finite nilpotent group. Consider the following (adjoint) identity from [BuPa, Theorem 9.6]: $$\left(\prod_{\chi \in \mathrm{Irr}(G)} \frac{\chi}{\chi(1)} \right)^2 =\frac{|Z(G)|}{|G|}\...
Sebastien Palcoux's user avatar
5 votes
0 answers
151 views

Finitely generated nilpotent groups with hyperbolic automorphisms

$\DeclareMathOperator\Out{Out}\DeclareMathOperator\GL{GL}$ Let $G$ be a finitely generated nilpotent group. We call an automorphism of $G$ hyperbolic if the induced automorphism of the free part of ...
Sean Lawton's user avatar
  • 8,384
5 votes
0 answers
170 views

Finitely generated nilpotent groups as cusp groups

I recently learned about the following question, asked by I. Kapovich : Is there an example of a group $G$ which is hyperbolic relative to some parabolic subgroups that are nilpotent of class $\geq 3$...
M. Dus's user avatar
  • 1,900
5 votes
0 answers
122 views

Regularity of polynomial growth of groups

Let $G$ be a finitely generated group of polynomial growth. This means that the size $B_n$ of the ball of radius $n$ satsifies: $$ A n^d \leq B_n \leq Bn^d $$ for some constants $A$, $B$. My question ...
Michal Kotowski's user avatar
4 votes
1 answer
320 views

Nilpotent of class 2 free product

Question. How is the nilpotent of class 2 (nil-2) free product of groups defined? I came across this construction reading the following paper. Alan H. Mekler (1981), Stability of nilpotent groups of ...
Filippo Calderoni's user avatar
4 votes
1 answer
791 views

On Canonical generators of torsion free nilpotent group

I read somewhere that Maltcev proved that for any finitely generated torsion free Nilpotent group $G$ there are canonical generators, i.e. $g_{1},\ldots,g_{k}$ such that any $g \in G$ can be written ...
Jonathan Hermon's user avatar
4 votes
0 answers
194 views

A different approach to proving a property of finite solvable groups

Edit: I'd be happy to hear any vague thoughts you might have, however far they may be from a complete solution! I asked this on math.stackexchange a couple of days ago, but it didn't attract any ...
semisimpleton's user avatar
4 votes
0 answers
132 views

Linear vs algebraic unipotent quotient stacks

Consider algebraic stacks of the form $\mathbb{C}^n/G$ where $G$ is a unipotent group satisfying either Type 1: $G$ acts on $\mathbb{C}^n$ via affine linear transformations Type 2: $G$ acts on $\...
Anton Mellit's user avatar
  • 3,572
4 votes
0 answers
136 views

Order problem in nilpotent groups

Let $G$ be a f.g. nilpotent group. I wanted to know if the order problem (given $g \in G$, deciding if there exists $n$ s.t. $g^n=e$) is decidable in $G$? In such a group, the word problem is ...
thibo's user avatar
  • 333
3 votes
2 answers
413 views

element of order n such that $\pi(n)=\pi(G)$, where $\pi(n)$ denote the prime divisors of $n$

Hello. I thank for your answer, in advance. Let $G$ be a finite group and $G$ has an element of order $n$ such that $\pi(n)=\pi(G)$ where $\pi(n)$ denote the set of prime divisors of $n$ and $\pi(G)$ ...
mousavi's user avatar
  • 81
3 votes
2 answers
326 views

Analyzing words in a "free" group of nilpotency class 2

Suppose that we have a group $G$ generated by a set $S$ of elements with the only family of relations being that all commutators are central. Equivalently, $G$ is the largest group of nilpotency ...
Inna's user avatar
  • 1,025
3 votes
2 answers
1k views

Are higher dimensional Heisenberg groups free nilpotent?

I know that the Heisenberg group { x,y | [x,[x,y]]=[y,[x,y]]=1 } is free nilpotent; what about the higher dimensional ones? Do the higer dimensional Heisenberg groups have nice presentations? By ...
dave's user avatar
  • 155
3 votes
1 answer
412 views

Stable homotopy of classifying space for nilpotent groups

Let $BG$ denote the classifying space of a (discrete) group and $BG_+$ its disjoint union with a point. Question: What is known about the stable homotopy groups $\pi^S_*(BG_+)$ ? If $G$ is finite (i....
user14120's user avatar
  • 347
3 votes
1 answer
113 views

Bounding size of group by number of generators, order of elements, and nilpotency class (Restricted Burnside's)

I have been looking at some papers on the "restricted Burnside problem". On page 4 of Vaughan-Lee and Zelmanov's survey, "Bounds in the restricted Burnside problem", I think they ...
Zach Hunter's user avatar
  • 3,393
3 votes
1 answer
133 views

Centre of solvable locally nilpotent groups

This question is motivated by two examples of locally nilpotent groups which I came across (see below). Question: Given an infinite solvable and locally nilpotent group $G$, does $G$ have an infinite ...
ARG's user avatar
  • 4,342
3 votes
1 answer
233 views

Free Lie algebra and nilpotent groups in Rothschild and Stein's paper

In Rothschild, Linda Preiss; Stein, Elias M., Hypoelliptic differential operators and nilpotent groups, Acta Math. 137(1976), 247-320 (1977). ZBL0346.35030. PDF at archive.ymsc.tsinghua.edu.cn ...
Houa's user avatar
  • 561
3 votes
1 answer
397 views

Maximal nilpotent subgroups of SO(n,1)

For the Lie group $SO(n,1)$ I believe the maximal nilpotent subgroups are conjugate to either a diagonal group times a compact group or a unipotent group times a compact group. In either case the ...
Davis's user avatar
  • 85