Finite Groups with Specific Subgroups Exhibiting Trivial Intersection Property

Abstract

This paper studies finite groups in which every Sylow p-subgroup has a trivial intersection with its conjugates, forming the class of TIP-groups. While classical conditions, such as the normality of Sylow p-subgroups, are equivalent to the group being nilpotent, the trivial intersection property defines a broader and more flexible class. We provide structural characterizations of TIP-groups and examine their relationship with well-known classes of finite groups. In particular, we relate TIP-groups to the p-cores by introducing the β-condition and offer examples to illustrate their distinguishing features.

Keywords: TIP-subgroups, Sylow subgroups, Nilpotent group, Solvable group.

2010 Mathematics Subject Classification. Primary 20D10, 20D20, 20D35