On Logically Cyclic Groups

TL;DR

This paper investigates the structure of logically cyclic groups, which are groups where a single element can define all others through first-order formulas, exploring their properties and classifications.

Contribution

It provides a detailed analysis of the structure of logically cyclic groups, offering new insights into their definability and classification within group theory.

Findings

Characterization of logically cyclic groups

Identification of conditions for definability

Structural classifications and properties

Abstract

A group $G$ is called logically cyclic, if it contains an element $s$ such that every element of $G$ can be defined by a first order formula with parameter $s$. The aim of this paper is to investigate the structure of such groups.