Finite Groups with Specific Subgroups Exhibiting Trivial Intersection Property

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