A characterization of nilpotent bicyclic groups
Kan Hu
TL;DR
The paper determines exactly when every $(m,n)$-bicyclic finite group is nilpotent, showing this holds if and only if $\\gcd(n,\\phi(\\operatorname{rad}(m)))=\\gcd(m,\\phi(\\operatorname{rad}(n)))=1$, extending the known abelian criterion $\\gcd(m,\\phi(n))=\\gcd(n,\\phi(m))=1$. The authors introduce the Mcond condition to separate nilpotent from non-nilpotent cases, construct explicit non-nilpotent examples when Mcond fails, and prove nilpotency via induction and Schur–Zassenhaus to obtain a direct product decomposition $G= P\times Q$. They further derive corollaries for abelian and cyclic instances, tying the nilpotent classification to singular-pair notions, and discuss related open problems and number-theoretic directions. The work solidifies the structural understanding of bicyclic groups and their role in related combinatorial and topological constructions, such as regular dessins and map classifications, with potential implications for counting problems in number theory.
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 $\gcd(m,φ(n))=\gcd(n,φ(m))=1$, where $φ$ 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 $\gcd(n,φ(\mathrm{rad}(m)))=\gcd(m,φ(\mathrm{rad}(n)))=1$, where $\mathrm{rad}(m)$ denotes the radical of $m$ (the product of its distinct prime divisors).