The infinite intersection property of groups

Articles in Press

Document Type : Research Paper

Author

Department of Sciences, Teacher Education College of Setif-Messaoud Zeghar, Setif, Algeria

Abstract

A group $G$ is said to satisfy the infinite trivial intersection property (ITIP for short), if for every pair of finite subgroups $U,V$ such that $U\cap V=1$, there exist infinite subgroups $X$ and $Y$ of $G$ such $U\leq X$ and $V\leq Y$ and $X\cap Y=1$. We shall say that a group $G$ satisfies the infinite non-trivial intersection property (INIP) if every pair of infinite subgroups of $G$ intersect non-trivially. The subject of this paper is to find classes of groups that satisfy ITIP. We prove, among other things, that every periodic locally nilpotent non-Chernikov group satisfies ITIP. The Pr\"{u}fer-by-finite p-groups are examples of locally nilpotent Chernikov groups that do not satisfy ITIP. We then characterize locally nilpotent groups that satisfy INIP and structure theorems are given in the periodic and the non-periodic case.

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References

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International Journal of Group Theory

Articles in Press, Corrected Proof
Available Online from 31 January 2025

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History

  • Receive Date: 23 September 2024
  • Revise Date: 26 January 2025
  • Accept Date: 31 January 2025
  • Published Online: 31 January 2025

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