The infinite intersection property of groups

Zitouni, Amel

doi:10.22108/ijgt.2025.142876.1927

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	The infinite intersection property of groups
International Journal of Group Theory
دوره 15، شماره 1، خرداد 2026، صفحه 1-8 اصل مقاله (415.42 K)
نوع مقاله: Research Paper
شناسه دیجیتال (DOI): 10.22108/ijgt.2025.142876.1927
نویسنده
Amel Zitouni^*
Department of Sciences, Teacher Education College of Setif-Messaoud Zeghar, Setif, Algeria
چکیده
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.
کلیدواژه‌ها
Locally nilpotent groups؛ Chernikov groups؛ infinite intersection property

مراجع
[1] A. O. Asar, Locally nilpotent p-groups whose proper subgroups are hypercentral or nilpotent-by-Chernikov, J. London Math. Soc. (2) 61 no. 2 (2000) 412–422. [2] C. J. B. Brookes and H. Heineken, Locally nilpotent groups with an intersection property, Arch. Math. (Basel), 44 no. 6 (1985) 488–492. [3] J. Hempel, The finitely generated intersection property for Kleinian groups, Knot theory and manifolds, Springer, Berlin, Heidelberg, (1985) 18–24. [4] S. R. Li and X. Y. Guo, Finite p-groups whose abelian subgroups have a trivial intersection, Acta Math. Sin. (Engl. Ser.) 23 no. 4 (2007) 731–734. [5] W. Möhres, Torsionsgruppen, deren Untergruppen alle subnormal sind, Geom. Dedicata, 31 no. 2 (1989) 237–244. [6] D. J. S. Robinson, Finiteness conditions and generalized soluble groups, Part I, II, Springer-Verlag, Berlin, 1972. [7] V. P. Shunkov, On locally finite groups with a minimality condition for Abelian subgroups, Algebr Logic, 9 (1970) 350–370. [8] A. I. Sozutov, Residually finite groups with nontrivial intersection of every subgroup pair, Siberian. Math. J., 41 no. 2 (2000) 362–365. [9] G. Walls, Trivial intersection groups, Arch. Math. (Basel), 32 no. 1 (1979) 1-4.
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The infinite intersection property of groups