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...

Let G be a connected reductive group defined over an algebraically closed field k of characteristic p > 0. The purpose of this paper is two-fold. First, when p is a good prime, we give a new proof of the “order formula” of D. Testerman for unipotent elements in G; moreover, we show that the same formula determines the p-nilpotence degree of the corresponding nilpotent elements in the Lie algebr...

In the first part of this paper we prove without using the transfer or characters the equivalence of some conditions, each of which would imply p-nilpotence of a finite group G. The implication of p-nilpotence also can be deduced without the transfer or characters if the group is p-constrained. For p-constrained groups we also prove an equivalent condition so that O ′ (G)P should be p-nilpotent...

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.

The method of Walker and Wood is used to completely determine the nilpotence height of the elements P s t in the Steenrod algebra at the prime 2. In particular, it is shown that (P s t ) 2bs/tc+2 = 0 for all s ≥ 0, t ≥ 1. In addition, several interesting relations in A are developed in order to carry out the proof.