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 $p^{th}$ …

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 …

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 …

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 …

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 …

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 grou …

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 tri …

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 $X$ i …

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 totie …

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. (O …

Let $G$ be a finitely generated nilpotent group. Then $G/[G,G]$ is finitely generated abelian group. Show that there exists a finite index subroup $H< G$ such that $H/[H,H]\sime …

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 p …

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 ha …