Large subgroup

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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]

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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 $H$ of a group $G$ is termed large in $G$ if for any subgroup $K$:

$H\cap K$ is trivial implies $K$ 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

• Conjugate-large subgroup
• Normality-large subgroup

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.
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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.
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ABOUT INTERMEDIATE SUBROUP CONDITION:View all properties satisfying intermediate subgroup condition | View facts about intermediate subgroup condition

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

In fact, the property of being large also satisfies the transfer condition. That is, if $H$ and $K$ are two subgroups of $G$ and $H$ is large in $G$, then $H\cap K$ is large in $K$.

Intersection-closedness

YES: This subgroup property is intersection-closed: an arbitrary (nonempty) intersection of subgroups with this property, also has this property.
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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.