The nilpotent-groups tag has no wiki summary.

**20**

votes

**2**answers

741 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 ...

**12**

votes

**0**answers

281 views

### 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 ...

**10**

votes

**0**answers

269 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 ...

**8**

votes

**1**answer

507 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 ...

**7**

votes

**1**answer

164 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 ...

**5**

votes

**5**answers

1k 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 ...

**5**

votes

**2**answers

231 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 ...

**4**

votes

**0**answers

101 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 ...

**4**

votes

**0**answers

367 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
...

**3**

votes

**2**answers

351 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 $\pi(G)$ ...

**3**

votes

**2**answers

808 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 ...

**3**

votes

**2**answers

235 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 ...

**3**

votes

**1**answer

170 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 ...

**3**

votes

**1**answer

224 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 ...

**3**

votes

**1**answer

153 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 ...

**2**

votes

**0**answers

81 views

### 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 ...

**1**

vote

**1**answer

275 views

### Homotopy classification of maps into nilmanifolds

I am interested in answers or reference in the literature to the following problem:
Classify up to homotopy all maps $A\to X$, where $A$ is a closed oriented manifold and $X$ is a closed ...

**1**

vote

**0**answers

100 views

### 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 ...