For $G$ a finite group, let $d_2(G)$ denote the proportion of triples $(x, y, z) \in G^3$ such that $[x, z] = 1$. We determine structure groups is bounded away from zero: if $d_2(G) \geq \epsilon > 0$, has class-4 nilpotent normal subgroup $H$ $[G : H] $ and $|\gamma_4(H)|$ are both in terms $\epsilon$. also show an infinite group whose commutators have boundedly many conjugates, or indeed sati...

We study multipliers on graded nilpotent Lie groups defined via group Fourier transform. More precisely, we show that Hörmander-type conditions the imply $L^p$-boundedness. express these using difference operators and

Mal’cev showed in the 1950s that there is a correspondence between radicable torsion-free nilpotent groups and rational nilpotent Lie algebras. In this paper we show how to establish the connection between the radicable hull of a finitely generated torsion-free nilpotent group and its corresponding Lie algebra algorithmically. We apply it to fast multiplication of elements of polycyclically pre...