Intermediately strictly characteristic 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]

Definition

Definition with symbols

A subgroup H of a group G is termed an intermediately strictly characteristic subgroup if for every subgroup K of G containing H, H is a strictly characteristic subgroup (also called distinguished subgroup) of K. Here, a strictly characteristic subgroup is a subgroup that is invariant under all surjective endomorphisms.

Relation with other properties

Stronger properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Intermediately fully invariant subgroup fully invariant in every intermediate subgroup
Homomorph-containing subgroup contains every homomorphic image in the whole group |FULL LIST, MORE INFO
Prehomomorph-contained subgroup contained in every subgroup having it as a homomorphic image
Normal subgroup having no nontrivial homomorphism to its quotient group no nontrivial homomorphism to its quotient group |FULL LIST, MORE INFO

Weaker properties

property quick description proof of implication proof of strictness (reverse implication failure) intermediate notions
Intermediately characteristic subgroup characteristic in every intermediate subgroup
Strictly characteristic subgroup
Characteristic subgroup