# Questions tagged [nilpotent-groups]

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### Rings or algebras with many nilpotent elements and efficient computation

Crossposted from quantum.SE where comment appears to suggest that solving modulo 2 might be possible. Searching the web for '"quantum computer" nilpotent' returns many results, so maybe the ...
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### 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/...
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### Nilpotency of topological groups

A group $G$ is said to be nilpotent if $G$ has a central series of finite length, that is, a series of normal subgroups $$\{1\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_n = G$$ ...
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### 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 ...
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### Descending FC series

In analogy to the central series one can define a FC series as a sequence $A_i$ of normal subgroups such that $$\{1\} = A_0 \lhd A_1 \lhd A_2 \lhd \cdots \lhd A_n = G$$ such that $A_{i+1}/ A_i$ is ...
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### 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 $\geq 3$...
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### 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 $N$ ...
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### 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 ...
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### Homeomorphism type of the horofunction boundary for nilpotent Lie groups

Consider a metric space $(X,d)$ and fix a base point $w$. A horofunction is a function of the form $$\beta_y(x)=d(x,y)-d(w,y).$$ The map $y\mapsto \beta_y$ is an embedding of $X$ into the space of $1$-...
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### (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). ...
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### 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 ...
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### 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....
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### infranilmanifolds: harmonic forms parallel?

I am studying Lott's paper : "On the spectrum of a finite volume negatively-curved manifold" and the satement is following: We have an compact infranilmanifold $N$ which is finitely covered by a ...
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### 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 ...
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### 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 $\pi(G)$ ...
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### 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 ...
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### 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 ...
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### 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 ...
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### Extending a representation to a finite little group

I have a question related to Mackey theory applied to discrete nilpotent groups which are not torsion-free and are infinite. Let us suppose that $G$ is a type I infinite discrete nilpotent group ...
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### 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 ...
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 ...
### 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 ...