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	Abelian hereditarily normal subgroup|FULL LIST, MORE INFO
SCAB-subgroup	every subgroup-conjugating automorphism of the whole group restricts to a subgroup-conjugating automorphism of the subgroup			Hereditarily normal subgroup|FULL LIST, MORE INFO
Abelian normal subgroup	abelian and a normal subgroup	abelian implies cyclic	cyclic not implies abelian	Homocyclic normal subgroup|FULL LIST, MORE INFO
Nilpotent normal subgroup	nilpotent and a normal subgroup	(via abelian)	(via abelian)	Abelian normal subgroup, Dedekind normal subgroup, Homocyclic normal subgroup|FULL LIST, MORE INFO
Solvable normal subgroup	solvable and a normal subgroup	(via abelian)	(via abelian)	Abelian normal subgroup, Dedekind normal subgroup, Nilpotent normal subgroup|FULL LIST, MORE INFO

Facts

• Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup