Conjugate-large subgroup

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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.

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 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.