Results on Engel Fuzzy Subgroups

Document Type : Research Paper


1 ‎Department of mathematics Payame Noor University P.O‎. ‎Box 19395-3697 Tehran‎, ‎Iran.

2 Department of mathematics Shahid Beheshti University‎, ‎G‎. ‎C.‎, ‎ Tehran‎, ‎Iran.

3 Department of Mathematics Education, Gyeong sang National university, Chinju, Korea



‎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.



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