# Large subgroup

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

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]

This subgroup property arises from a property of elements in lattices, when applied to the given subgroup as an element in the lattice of subgroups of a given group.

*You might be looking for:* large abelian subgroup, equivalent to being an abelian subgroup of maximum order in a group of prime power order

## Definition

### Symbol-free definition

A subgroup of a group is termed large if any subgroup that intersects it trivially must be the trivial subgroup.

### Definition with symbols

A subgroup of a group is termed **large** in if for any subgroup :

is trivial implies is trivial

### In terms of the large operator

The subgroup property of being large is the result of applying the large operator to the tautology subgroup property, that is, the property of being *any* subgroup.

## Relation with other properties

### Weaker properties

## Metaproperties

### Transitivity

This subgroup property is transitive: a subgroup with this property in a subgroup with this property, also has this property in the whole group.ABOUT THIS PROPERTY: View variations of this property that are transitive | View variations of this property that are not transitiveABOUT TRANSITIVITY: View a complete list of transitive subgroup properties|View a complete list of facts related to transitivity of subgroup properties |Read a survey article on proving transitivity

A large subgroup of a large subgroup is large. This follows from the general theory of the large operator.

### Intermediate subgroup condition

YES:This subgroup property satisfies the intermediate subgroup condition: if a subgroup has the property in the whole group, it has the property in every intermediate subgroup.ABOUT THIS PROPERTY: View variations of this property satisfying intermediate subgroup condition | View variations of this property not satisfying intermediate subgroup conditionABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

If is a large subgroup of , is also a large subgroup of any intermediate subgroup of containing . This follows directly from the definition.

In fact, the property of being large also satisfies the transfer condition. That is, if and are two subgroups of and is large in , then is large in .

### 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-closedABOUT 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

The property of being large is finite-intersection-closed. The fact that it is finite-intersection-closed follows from its being transitive and satisfying the transfer condition.

### Upward-closedness

This subgroup property is upward-closed: if a subgroup satisfies the property in the whole group, every intermediate subgroup also satisfies the property in the whole group

View other upward-closed subgroup properties

Any subgroup containing a large subgroup is large. This again follows directly from the definition.