in this paper we study right $n$-engel group elements‎. ‎by modifying a group constructed by newman and nickel‎, ‎we construct‎, ‎for each integer $ngeq 5$‎, ‎a 2-generator group $g =langle a‎, ‎brangle$ with the property that $b$ is a right $n$-engel‎ ‎element but where $[b^k,_n a]$ is of infinite order when $knotin {0‎, ‎1}$‎.

‎In the classical group theory there is‎ an open question‎: ‎Is every torsion free n-Engel group (for n ≥ 4)‎, nilpotent?‎. ‎To answer the question‎, ‎Traustason‎ [11] showed that with some additional conditions all‎ ‎4-Engel groups are locally nilpotent‎. ‎Here‎, ‎we gave some partial‎ answer to this question on Engel fuzzy subgroups‎. ‎We show that if μ is a normal 4-Engel fuzzy‎ subgroup of ...

A relationship between two natural dynamical systems on groups is established thereby giving a new characterization of Engel elements. Using this connection, various closure properties for the sets of left and right Engel elements are established. Some other dynamical systems and their relationship with Engel elements and nilpotency for finite groups are also considered.

In this paper we study left and right 4-Engel elements of a group. In particular, we prove that 〈a, a〉 is nilpotent of class at most 4, whenever a is any element and b are right 4-Engel elements or a are left 4-Engel elements and b is an arbitrary element of G. Furthermore we prove that for any prime p and any element a of finite p-power order in a group G such that a ∈ L4(G), a, if p = 2, and ...

In this paper we study left 3-Engel elements in groups. In particular, we prove that for any prime p and any left 3-Engel element x of finite p-power order in a group G, x is in the Baer radical of G. Also it is proved that 〈x, y〉 is nilpotent of class 4 for every two left 3-Engel elements in a group G.