Image-closed characteristic subgroup

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Definition

Symbol-free definition

A subgroup of a group is termed an image-closed characteristic subgroup if, under any surjective homomorphism, its image is a characteristic subgroup of the image.

Definition with symbols

A subgroup $H$ of a group $G$ is termed image-closed characteristic in $G$ if, for any normal subgroup $N$ of $G$, $HN/N$ is a characteristic subgroup of $G/N$.

Relation with other properties

Stronger properties

• Verbal subgroup

Weaker properties

• Characteristic subgroup
• Normal subgroup