Conjugate-comparable subgroup
From Groupprops
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
Contents
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 |
