# Conjugate-large subgroup

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]

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

## Definition

### 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.

## 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 transitive
ABOUT 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

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 identity-true subgroup property, but it is not in general trivially true.