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

Document Type : Research Paper


1 Department of Pure Mathematics, Ferdowsi University of Mashhad, Mashhad, Iran.

2 Department of Mathematics, Quchan University of Technology, Quchan, Iran.


‎Let G be a finite group. The notion of ‎‎‎n‎-Engel degree of G,‎ denote 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. ‎‎‎


  1. A. M. A. Alghamdi, F. G. Russo, A generalization of the probability that the commu[1]tator of two group elements is equal to a given element, Bull. Iranian Math. Soc., 38, 973–986 (2012).
  2. N. M. M. Ali, N. H. Sarmin, On some problems in group theory of probabilistic nature, Menemui Matematik, 32, 35–41 (2010).
  3. R. Barzgar, A. Erfanian, M. Farrokhi D. G., Finite groups with three relative commu[1]tativity degrees, Bull. Iranian Math. Soc., 39, 271–280 (2013).
  4. P. G. Crosby, G. Traustason, On right n-Engel subgroups, J. Algebra, 324, 875–883 (2010).
  5. P. Erdös, P. Turán, On some problems of statistical group-theory. IV, Acta Math. Acad. Sci. Hung., 19, 413–435 (1968).
  6. A. Erfanian, M. Farrokhi D. G., On the probability of being a 2-Engel groups, Int. J. Group Theory, 2, 31–38 (2013).
  7. A. Erfanian, R. Rezaei, P. Lescot, On the relative commutativity degree of a subgroup of a finite group, Comm. Algebra, 35, 4183–4197 (2007).
  8. R. M. Guralnich, G. R. Robinson, On the commuting probability in finite groups, J. Algebra, 300, 509–528 (2006).
  9. P. Lescot, Isoclinism classes and commutativity degrees of finite goups, J. Algebra, 177, 847–869 (1995).
  10. W. P. Kappe, Die A–norm einer gruppe, Illinois. J. Math., 5, 187–197 (1961).