Conjugate-comparable subgroup

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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

• Any finite subgroup that is conjugate-comparable is normal.
• Any subgroup of finite index that is conjugate-comparable is normal.
• Any subgroup of a slender group, Artinian group, or periodic group that is conjugate-comparable is normal.

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

• Automorph-comparable of normal implies conjugate-comparable
• Left residual of conjugate-comparable by normal is automorph-comparable