# 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