On the Relative 2-Engel Degree of A Subgroup of A Finite Group

Let G be a finite group. The notion of n-Engel degree of G, denoted by dn(G), is the probability of two randomely chosen elements x, y ∈ G satisfy the n-Engel condition [y,n x] = 1. The case n = 1 is the known commutativity degree of G. The aim of this paper, is to define and investigate the relative 2-Engel degree of a subgroup H of G as the probability of two randomely chosen elements x ∈ G and y ∈ H satisfy the 2-Engel condition [y,2 x] = 1.