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]

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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.

Relation with other properties

Stronger properties

• Large subgroup

Weaker properties

• 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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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.