Results on Engel Fuzzy Subgroups

Document Type : Research Paper

Authors

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

10.29252/asta.4.2.1

Abstract

‎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.

 

Keywords


[1] R. Ameri, R.A. Borzooei, E. Mohammadzadeh, Engel fuzzy subgroups, Italian Journal of Pure and Applied Mathemetics, 34 (2015), pp. 251-262.
[2] R.A. Borzooei, M. Bakhshi, T-fuzzy congruences and T-fuzzy filters of a BL-algebra, Iranian Journal of Fuzzy Systems, 6(4) (2009), pp. 37-47. 
[3] R.A. Borzooei, E. Mohammadzadeh, V. Fotea, On Engel Fuzzy Subpolygroups, New Mathematics and Natural Computation, to appear. 
[4] E.S. Golod, Some problems of Burnside type, Proc. Int. Congr. Math., Moscow (1966), pp. 284-289. 
[5] H. Heineken, Engelsche Elemente der Lange drei, Illinois Journal of Mathematics, 5 ( 1961), pp. 681-707. 
[6] L.C. Kappe and W.P. Kappe, On three-Engle groups, Bulletin of the Australian Mathematical Society, 7(3) (1972), pp. 391-405. 
[7] F.W. Levi, Groups in which the commutator operation satiesfi es certain algebric conditions, Journal of the Indian Mathematical Society, 6 (1942), pp. 87-97. 
[8] J.N. Mordeson, K.R. Bhutani, A. Rosenfeld, Fuzzy group theory, Springer (2005). 
[9] D. Robinson, A course in the theory of groups, Springer (2012).
[10] A. Rosenfeld, Fuzzy groups, Journal of Mathematical Analysis and Applications, 35(3) (1971), pp. 512-517. 
[11] G.Traustason, On 4-Engle groups, Journal of Algebra, 178(2) (1995), pp. 414-429. 
[12] J. Wiegold, Transitivegroups with fixed-point-free permutations, Archiv der Mathematik, 27 (1976), pp. 473-475. 
[13] J. Wiegold, Transitivegroups with fixed-point-free permutations II, Archiv der Mathematik, 29 (1977), pp. 571-573. 
[14] L.A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), pp. 338-353. 
[15] M. Zorn, Nilpotency of finite groups, Bulletin of the American Mathematical Society, 42 (1936), pp. 485-486. 
[16] A. Abdollahi, S. Janbaz and M. Oubodi, Graphs Cospectral with A Friendship Graph Or its Complement, Trans. Combin. 2(4) (2013), pp. 37-52. 
[17] N.L. Biggs, Algebraic Graph Theory, (second edition), Cambridge University press, cambridge, (1933).