Upward-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

A subgroup $H$ of a group $G$ is termed upward-closed characteristic if, for every subgroup $K$ of $G$ containing $H$, $K$ is also a characteristic subgroup of $G$.

Formalisms

In terms of the upward-closure operator

This property is obtained by applying the upward-closure operator to the property: characteristic subgroup
View other properties obtained by applying the upward-closure operator

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Cyclic-quotient characteristic subgroup	characteristic subgroup, quotient group is cyclic	cyclic-quotient characteristic implies upward-closed characteristic	upward-closed characteristic not implies cyclic-quotient in finite	|FULL LIST, MORE INFO
Characteristic subgroup of prime index	characteristic subgroup and subgroup of prime index			|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
Upward-closed normal subgroup	every subgroup containing it is normal in whole group