Quotient-isomorph-containing 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 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 \le 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