Groups with Ordered Structures

An Engel condition for orderable groups

“…In particular, every residually finite group is locally graded. Kim and Rhemtulla also proved that any orderable n-Engel group is nilpotent ( [5], see also [6]). A group G is called orderable if there exists a full order relation ≤ on the set G such that x ≤ y implies axb ≤ ayb for all a, b, x, y ∈ G, i.e.…”

On some classes of orderable groups

“…Ordered k-Engel groups have been studied by Y. Kim and A. Rhemtulla in [16]. They proved the following…”

“…Furthemore we survey results about ordered and right ordered n-Engel groups. We show that an n-Engel O-group is nilpotent (see [16]), and that an RO 4-Engel group satisfies a non-trivial semigroup law and then it is nilpotent. We point out that it is still an open question whether an RO n-Engel group is nilpotent.…”

“…Ordered and right ordered groups have been considered by many authors (see for instance [2,4,10,16,19,20]). …”

Engel conditions on orderable groups and in combinatorial problems (a survey)

“…Theorem (P. L., M. Maj and A.H. Rhemtulla, 1995) Let V is the variety of all nilpotent groups of class at most k defined by the law [y 1 , . .…”