# Intersection of Sylow subgroups

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## 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.

## Metaproperties

### Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
ABOUT THIS PROPERTY: View variations of this property that are intersection-closed | View variations of this property that are not intersection-closed
ABOUT INTERSECTION-CLOSEDNESS: View all intersection-closed subgroup properties (or, strongly intersection-closed properties) | View all subgroup properties that are not intersection-closed | Read a survey article on proving intersection-closedness | Read a survey article on disproving intersection-closedness