Intersection of Sylow subgroups

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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.

Relation with other properties

Stronger properties

Weaker properties

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