Large subgroup

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.

Weaker properties

 * Conjugate-large subgroup
 * Normality-large subgroup

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

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 $$K$$.

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.

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