Quotient-isomorph-containing subgroup

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 $G$ is a group and $H$ is a subgroup. We say that $H$ is a quotient-isomorph-containing subgroup of $G$ if the following is true: $H$ is a normal subgroup of $G$, and if $K$ is a normal subgroup of $G$ such that the quotient groups $G/H$ and $G/K$ are isomorphic groups, then $H\leq K$.

If $G$ is a finite group, this property is equivalent to being a quotient-isomorph-free subgroup.

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
quotient-isomorph-free subgroup	normal and no other normal subgroup has an isomorphic quotient group.			|FULL LIST, MORE INFO
quotient-homomorph-containing subgroup				|FULL LIST, MORE INFO

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
characteristic subgroup	no other automorphic subgroup.			Template:Intermediate notion short

Related properties

• Isomorph-containing subgroup is a subgroup that contains every isomorphic subgroup to it in the whole group.