An Arithmetic Characterization of 2-Generated Numbers

TL;DR

This paper provides an arithmetic characterization of 2-generated numbers, identifying when all groups of a given order are generated by two elements based on their prime factorization.

Contribution

It introduces a novel arithmetic criterion for determining 2-generated numbers using prime factorization, advancing understanding of group generation properties.

Findings

Characterization of 2-generated numbers via prime factorization

Criteria for all groups of order n to be 2-generated

Extension of group generation theory to arithmetic conditions

Abstract

A group $G$ is said to be $k$-generated if it has a generating set with $k$ elements. A positive integer $n$ is called a \emph{2-generated number} if every group of order $n$ is 2-generated. In this article, we establish an arithmetic characterization of 2-generated numbers expressed in terms of the prime factorization of $n$.