Image-potentially fully invariant 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

Suppose $H$ is a subgroup of a group $G$. We say that $H$ is an image-potentially fully invariant subgroup of $G$ if there exists a group $K$, a surjective homomorphism $\rho:K \to G$, and a subgroup $L$ of $K$ such that $\rho(L) = H$.

Relation with other properties

Stronger properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Fully invariant subgroup

Weaker properties

property	quick description	proof of implication	proof of strictness (reverse implication failure)	intermediate notions
Normal subgroup