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]
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

Relation with other properties

Stronger properties

Weaker properties

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.