The Probability That an Ordered Pair of Elements is an Engel Pair

Amiri, S.M. Jafarian
Rostami, Hojjat

Journal of the Indonesian Mathematical Society
Volume 25 Number 2 (July 2019)

Let G be a nite group. We denote by ep(G) the probability that[x;n y] = 1 for two randomly chosen elements x and y of G and some posi-tive integer n. For x 2 G we denote by EG(x) the subset fy 2 G : [y;n x] =1 for some integer ng. G is called an E-group if EG(x) is a subgroup of G for allx 2 G. Among other results, we prove that if G is an non-abelian E-group withep(G) 16 , then G is not simple and minimal non-solvable.