Generalization of order separability for free products and omnipotence of free products of groups

TL;DR

This paper extends order separability results to free products of residually finite groups, showing that elements can be mapped onto finite groups with prescribed order multiples, highlighting a form of omnipotence.

Contribution

It generalizes order separability to broader classes of free products, demonstrating omnipotence properties for elements under specific conditions.

Findings

Existence of homomorphisms onto finite groups with prescribed element orders

Extension of order separability to free products of residually finite groups

Demonstration of omnipotence in free products

Abstract

It was proved that for any finite set of elements of a free product of residually finite groups such that no two of them belong to conjugate cyclic subgroups and each of them do not belong to a subgroup which is conjugate a to free factor there exists a homomorphism of the free product onto a finite group such that the order of the image of each fixed element is an arbitrary multiple of a constant number.