Strongly embedded 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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This article is about a term related to the Classification of finite simple groups

Definition

Symbol-free definition

A subgroup of a group is termed strongly embedded or tightly embedded if it has even order, is self-normalizing and its intersection with any other conjugate has odd order.

Definition with symbols

A subgroup $H$ of a group $G$ is termed strongly embedded or tightly embedded in $G$ if $H$ has even order and for any $x$ in $G$ which is not in $H$ , $H$ ∩ $H^{x}$ has odd order.

Relation with other 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.
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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

The condition of having even order is clearly transitive, while the condition of the intersection with any conjugate having odd order is also transitive. For full proof, refer: Strongly embedded satisfies transitivity