On Positive Law Problems in the Class of Locally Graded Groups

Abstract

We consider five known problems, show that three of them are equivalent, and all of them have an affirmative answer in the class of locally graded groups. Outside of this class a counterexample is known to only one problem.

Positive Laws on Large Sets of Generators: Counterexamples for Infinitely Generated Groups

“…However, the answer is positive for a large class of groups: Burns and Medvedev proved in [2] that a locally graded group satisfying a positive law is nilpotent-by-(locally finite of finite exponent). (See also the paper [1] by Bajorska and Macedońska. ) An interesting question regarding positive laws is the following: under what conditions does a positive law on a set T of generators of a group G imply a (possibly different) positive law on the whole of G?…”

On difficult problems and locally graded groups

“…Proof It is shown in [2], that every collapsing LG-group satisfies a positive law, which implies by Theorem 2, that G is nilpotent-by-'locally finite of finite exponent', and proves the "if" part. The "only if" part follows because nilpotent-by-'locally finite of finite exponent' groups are LG-groups.…”