Conjugate-large subgroup

Symbol-free definition
A subgroup of a group is said to be conjugate-large if any subgroup all of whose conjugates intersect it trivially must be the trivial subgroup.

Definition with symbols
A subgroup $$H$$ of a group $$G$$ is said to be conjugate-large if whenever $$K$$ is a subgroup such that $$K^g \cap H$$ is trivial for every $$g \in G$$, $$K$$ must itself be trivial.

Stronger properties

 * Large subgroup

Weaker properties

 * Normality-large subgroup

Metaproperties
We need to show that if $$H \le K \le G$$ with each conjugate-large in the next, $$H$$ is conjugate-large in $$G$$.

The proof of this is as follows: let $$M \le G$$ such that $$M^g \cap K$$ is trivial. We first show that $$M \cap K$$ is trivial by observing that $$H$$ is conjugate-large in $$K$$. We then show that $$M$$ is trivial using the fact that $$K$$ is conjugate-large in $$G$$.

Trimness
Conjugate-largeness is an, but it is not in general trivially true.