The ‘degree of k-step nilpotence’ a finite group G is the proportion tuples (x1,…, xk+1 ∈ Gk+1 for which simple commutator [x1, …, xk+1] equal to identity. In this paper we study versions an infinite G, with degree nilpotence defined by sampling in various natural ways, such as random walk, or Følner sequence if amenable. our first main result show that finitely generated, then positive and onl...

An MC group is a group in which all chains of centralizers have finite length. In this article, we show that every nilpotent subgroup of an MC group is contained in a definable subgroup which is nilpotent of the same nilpotence class. Definitions are uniform when the lengths of chains are bounded.

I show that simple finite vertex algebras are commutative, and that the Lie conformal algebra structure underlying a reduced (= without nilpotent elements) finite vertex algebra is nilpotent.

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x; δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about ...

In this paper, we focus on Hall's criterion for nilpotence in semi-abelian categories, and improve the bound of main theorem [2, Theorem 3.4] (see Main Theorem). And is best possible.