AI 中文总结
该研究针对有限亚循环p群，细化了可分与不可分亚循环群的概念，通过|K|刻画了亚循环结构的不同类型，并确定了这些类型在同一群上的共存情况。
AI 中文摘要
设P为有限亚循环p群，其中p为奇素数。我们对可分亚循环群与不可分亚循环群的概念进行了细化，区分亚循环结构（即P的循环正规子群K，满足P/K也为循环群）是不可分、可分还是自动可分的。我们证明这些不同类型可通过|K|来刻画，还确定了哪些类型可在同一群P上共存。
英文摘要
Let $P$ be a finite metacyclic $p$-group where $p$ is an odd prime. We refine the notions split metacyclic and non-split metacyclic group by distinguishing whether a metacyclic structure (i.e. a cyclic normal subgroup $K$ of $P$ such that $P/K$ is also cyclic) is non-splittable or splittable or automatically split. We show that these different types can be characterised in terms of $|K|$. We also determine which of these types can coexist on the same group $P$.
Comments12 pages
