SOME COMMUTATOR GROUP LAWS EQUIVALENT TO THE COMMUTATIVE LAW

ABSTRACT

Let G be a group, n an integer greater than 2 and . Let be a commutator of weight n with entries from the set . The question, originally raised by N. D. Gupta, is: Under what conditions is a group satisfying the law abelian? Some sufficient conditions for an affirmative answer are the finiteness or the solvability of the group. It is known that in the case the answer is positive for an arbitrary group. We prove that every group satisfying the law is abelian. We establish some group laws of the form which imply the commutative law without any other restriction. We also prove that every 4-Engel group satisfying the law is abelian.

ACKNOWLEDGMENT

Supported by the Ministry of Science and Technology of Slovenia.