Logical generation of groups

Abstract

A group G is called logically generated by a subset S, if every element of G can be defined by a first order formula with parameters from S. We consider the case where G is a direct product of finite nilpotent groups with mutually coprime orders and we show that logical and algebraic generations are equivalent in G. We also prove that in the case when G is a free non-abelian group, if S logically generates G then either it generates G algebraically or $〈S〉$ is not a free factor of G.

KEYWORDS:

• Definability
• elementary extensions
• logically cyclic groups
• logical generation

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 20A15
• Secondary: 03C07
• 03C98

Acknowledgments

We would like to thank the referee for his/her comments and suggestions.