Transfer-closed characteristic 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]

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Definition

Definition with symbols

A subgroup $H$ of a group $G$ is termed a transfer-closed characteristic subgroup if, for any subgroup $K \le G$, $H \cap K$ is a characteristic subgroup of $K$.

Formalisms

In terms of the transfer condition operator

This property is obtained by applying the transfer condition operator to the property: characteristic subgroup
View other properties obtained by applying the transfer condition operator

Relation with other properties

Stronger properties

• Normal Sylow subgroup
• Normal Hall subgroup
• Variety-containing subgroup
• Subisomorph-containing subgroup

Weaker properties

• Intermediately characteristic subgroup
• Characteristic subgroup

Metaproperties

Metaproperty name	Satisfied?	Proof	Difficulty level	Statement with symbols
transitive subgroup property	Yes	transfer-closed characteristicity is transitive		If $H \le K \le G$ are groups such that $H$ is transfer-closed characteristic in $K$ and $K$ is transfer-closed characteristic in $G$, then $H$ is transfer-closed characteristic in $G$.
intermediate subgroup condition	Yes			If $H \le K \le G$ are groups such that $H$ is transfer-closed characteristic in $G$, then $H$ is transfer-closed characteristic in $K$.
strongly intersection-closed subgroup property	Yes			If $H_i, i \in I$ is a family of transfer-closed characteristic subgroups of a group $G$, the intersection $\bigcap_{i \in I} H_i$ is also a transfer-closed characteristic subgroup of $G$.
transfer condition	Yes			If $H$ and $K$ are subgroups of $G$ such that $H$ is transfer-closed characteristic in $G$, then $H \cap K$ is transfer-closed characteristic in $K$.