# Conjugate-large subgroup

From Groupprops

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 of a group is said to be **conjugate-large** if whenever is a subgroup such that is trivial for every , must itself be trivial.

## Relation with other properties

### Stronger 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

We need to show that if with each conjugate-large in the next, is conjugate-large in .

The proof of this is as follows: let such that is trivial. We first show that is trivial by observing that is conjugate-large in . We then show that is trivial using the fact that is conjugate-large in .

### Trimness

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