2025/05/08 by Kan Hu, Hu, Kan
Engineering · Mathematics · #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #Rings, Modules, and Algebras #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.2505.05065
openalex publication_date 2025/05/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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(rad(m)))=gcd(m,ϕ(rad(n)))=1, where rad(m) denotes the radical of m (the product of its distinct prime divisors).