This paper introduces the concept of auto–ENGEL polygroups via the heart of hypergroups and investigates the relation between of auto–ENGEL polygroups and auto–nilpotent polygroups. Indeed, we show that the concept of heart of hypergroups plays an important role on construction of auto–ENGEL polygroups. This study considers the notation of characteristic set in hypergroups with respect to automorphism of hypergroups and shows that the heart of hypergroups is a characteristic set in hypergroups.

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 study some inverse problems of small doubling type in the class of m-ENGEL groups. In particular we investigate the structure of a , nite subset S of a torsion-free m-ENGEL group if j S = 2 j S j + b, where 0 ,b ,j S j 4, for some values of b.

The eighth edition of the international series of Groups St Andrews conferences was held at the University of Bath in 2009 and one of the theme days was dedicated to ENGEL groups. Since then much attention has been devoted to a verbal generalization of ENGEL groups. In this paper we will survey the development of this investigation during the last decade.

We survey left 3-ENGEL elements in groups.

The theory of ENGEL groups plays an important role in group theory since these groups are closely related to the Burnside problems. In this survey we consider several classical and novel algorithmic problems for ENGEL groups and propose several open problems. We study these problems with a view towards applications to cryptography.