Intersection of Sylow subgroups

Definition
A subgroup of a finite group is termed an intersection of Sylow subgroups if it can be expressed as an intersection of Sylow subgroups of the whole group.

Note that if we intersect $$p$$-Sylow subgroups for different primes $$p$$, then we get the trivial subgroup. Thus, any nontrivial subgroup obtained as an intersection of Sylow subgroups is obtained as an intersection of $$p$$-Sylow subgroups for a single prime $$p$$. We're often interested in studying only intersections of $$p$$-Sylow subgroups for a specified prime $$p$$.

The related term Sylow intersection is typically used for a subgroup obtained as an intersection of two distinct $$p$$-Sylow subgroups.

Stronger properties

 * Weaker than::Maximal Sylow intersection
 * Weaker than::Tame Sylow intersection
 * Weaker than::Sylow intersection
 * Weaker than::Sylow subgroup
 * Weaker than::Sylow-core

Weaker properties

 * Stronger than::Core-characteristic subgroup