# Cyclic normal subgroup

From Groupprops

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

## Contents

## 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 groupVIEW: 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 |