Finite direct power-closed characteristic subgroup

Definition

Definition with symbols

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is finite direct power-closed characteristic in $G$ if the following holds for every natural number $n$: in the group $G^n$ which is defined as the external direct product of $n$ copies of $G$, the corresponding subgroup $H^n$ is a 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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Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
fully invariant subgroup	invariant under all endomorphisms	combine full invariance is finite direct power-closed and fully invariant implies characteristic	finite direct power-closed characteristic not implies fully invariant	|FULL LIST, MORE INFO
bound-word subgroup	given by a system of conditions with quantifiers on all other variables			|FULL LIST, MORE INFO
verbal subgroup				Fully invariant subgroup|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	invariant under all automorphisms	(by definition)	characteristicity is not finite direct power-closed	|FULL LIST, MORE INFO
normal subgroup	invariant under all inner automorphisms	(via characteristic)	(via characteristic)	Characteristic subgroup|FULL LIST, MORE INFO