Cyclic-quotient characteristic subgroup
Definition
A subgroup of a group is termed a cyclic-quotient characteristic subgroup if it is a characteristic subgroup and the quotient group is a cyclic group.
Note that it makes sense to talk of the quotient group for a characteristic subgroup because any characteristic subgroup is normal.
Relation with other properties
Stronger properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Characteristic subgroup of prime index |
Weaker properties
| Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
|---|---|---|---|---|
| Upward-closed characteristic subgroup | every subgroup containing it is characteristic | cyclic-quotient characteristic implies upward-closed characteristic | upward-closed characteristic not implies cyclic-quotient in finite | |
| Characteristic subgroup | |FULL LIST, MORE INFO | |||
| Cyclic-quotient subgroup | quotient group is cyclic | |||
| Abelian-quotient characteristic subgroup | characteristic and quotient group is abelian | |||
| Abelian-quotient subgroup | quotient group is abelian | |FULL LIST, MORE INFO |