A characterization of nilpotent bicyclic groups

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).