# Transfer-closed characteristic subgroup

BEWARE! This term is nonstandard and is being used locally within the wiki. [SHOW MORE]
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]
This is a variation of characteristicity|Find other variations of characteristicity | Read a survey article on varying characteristicity

## 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

## 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 $K$.