The infinite intersection property of groups

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=۱, there exist infinite subgroups X and Y of G such U\leq X and V\leq Y and X\cap Y=۱. 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.

کلیدواژه ها:

locally nilpotent groups ، Chernikov groups ، infinite intersection property