Cyclic normal subgroup
This article describes a property that arises as the conjunction of a subgroup property: normal subgroup with a group property (itself viewed as a subgroup property): cyclic group
View a complete list of such conjunctions
Definition
Symbol-free definition
A subgroup of a group is termed a cyclic normal subgroup if it is cyclic as a group and normal as a subgroup.
Examples
VIEW: subgroups satisfying this property | subgroups dissatisfying property normal subgroup | subgroups dissatisfying property cyclic group
VIEW: Related subgroup property satisfactions | Related subgroup property dissatisfactions
Relation with other properties
Stronger properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Cyclic characteristic subgroup | cyclic and a characteristic subgroup | characteristic implies normal | normal not implies characteristic (e.g., cyclic subgroup of Klein four-group) | |FULL LIST, MORE INFO |
Weaker properties
| property | quick description | proof of implication | proof of strictness (reverse implication failure) | intermediate notions |
|---|---|---|---|---|
| Hereditarily normal subgroup | cyclic normal implies hereditarily normal | hereditarily normal not implies cyclic normal | |FULL LIST, MORE INFO | |
| SCAB-subgroup | every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup | |FULL LIST, MORE INFO | ||
| Abelian normal subgroup | abelian and a normal subgroup | cyclic implies abelian | abelian not implies cyclic | |FULL LIST, MORE INFO |
| Nilpotent normal subgroup | nilpotent and a normal subgroup | (via abelian) | (via abelian) | Abelian normal subgroup|FULL LIST, MORE INFO |
| Solvable normal subgroup | solvable and a normal subgroup | (via abelian) | (via abelian) | Abelian normal subgroup|FULL LIST, MORE INFO |