Conjugate-comparable 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]
This is a variation of normal subgroup|Find other variations of normal subgroup | Read a survey article on varying normal subgroup

Definition

A subgroup of a group is termed a conjugate-comparable subgroup if it is comparable with each of its conjugate subgroups, in other words, every conjugate subgroup to it either contains it or is contained in it.

Relation with other properties

Collapse to normality

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Automorph-comparable subgroup comparable to all its automorphic subgroups |FULL LIST, MORE INFO
Normal subgroup equal to each conjuate subgroup equal things can be compared conjugate-comparable not implies normal |FULL LIST, MORE INFO

Weaker properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
Subgroup invariant under conjugation by a generating set

Facts