Results on Engel Fuzzy Subgroups

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 group G, x,y in G and a =yx, then μ|< a, y> is a generalized nilpotent of class at most 2. Also we define a torsion free fuzzy subgroup and show that if μ is a 4-Engel torsion free fuzzy subgroup of G, then μ|< a, y> is a generalized nilpotent of class at most 4, for conjugate elements a,y in G.