A characterization of nilpotent bicyclic groups

TL;DR

This paper characterizes when finite $(m,n)$-bicyclic groups are nilpotent, extending previous results on abelian groups by involving gcd conditions with Euler's totient and the radical of integers.

Contribution

It generalizes the classification of finite $(m,n)$-bicyclic groups from abelian to nilpotent by establishing gcd conditions involving Euler's totient and the radical of integers.

Findings

Finite $(m,n)$-bicyclic groups are nilpotent if and only if specific gcd conditions hold.

The gcd conditions involve Euler's totient function and the radical of the integers $m$ and $n$.

This extends previous results characterizing abelian bicyclic groups.

Abstract

A group is called $(m,n)$-bicyclic if it can be expressed as a product of two cyclic subgroups of orders $m$ and $n$, respectively. The classification and characterization of finite bicyclic groups have long been important problems in group theory, with applications extending to symmetric embeddings of the complete bipartite graphs. A classical result by Douglas establishes that every bicyclic group is supersolvable. More recently, Fan and Li (2018) proved that every finite $(m,n)$-bicyclic group is abelian if and only if $\gcd(m,\phi(n))=\gcd(n,\phi(m))=1$, where $\phi$ is Euler's totient function. In this paper we generalize this result further and show that every $(m,n)$-bicyclic group is nilpotent if and only if $\gcd(n,\phi(\mathrm{rad}(m)))=\gcd(m,\phi(\mathrm{rad}(n)))=1$, where $\mathrm{rad}(m)$ denotes the radical of $m$ (the product of its distinct prime divisors).