Weakly image-closed fully invariant subgroup

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Definition

A subgroup $H$ of a group $G$ is termed a weakly image-closed fully invariant subgroup if, for any normal subgroup $N$ of $G$ contained in $H$, $H/N$ is a fully invariant subgroup of $G/N$.

Relation with other properties

Stronger properties

• Image-closed fully invariant subgroup
• Quotient-subisomorph-containing subgroup
• Verbal subgroup

Weaker properties

• Fully invariant subgroup
• Weakly image-closed characteristic subgroup
• Characteristic subgroup
• Normal subgroup