# Group in which every nontrivial normal subgroup contains a cyclic normal subgroup

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

A **group in which every nontrivial normal subgroup contains a cyclic normal subgroup** is a group with the property that any nontrivial normal subgroup of the group, contains a cyclic normal subgroup.

## Relation with other properties

### Stronger properties

- Abelian group
- Nilpotent group:
`For full proof, refer: Nilpotent implies every nontrivial normal subgroup contains a cyclic normal subgroup` - Group whose center is normality-large:
`For full proof, refer: Center is normality-large implies every nontrivial normal subgroup contains a cyclic normal subgroup` - Supersolvable group:
`For full proof, refer: Supersolvable implies every nontrivial normal subgroup contains a cyclic normal subgroup`

## Metaproperties

### Quotients

This group property isnotquotient-closed, viz., we could have a group with the property and a quotient group of that group that doesnothave the property

It can happen that a group satisfies the property that every nontrivial normal subgroup contains a cyclic normal subgroup, but the quotient of doesn't satisfy that property. For instance, if we take to be a quasisimple group that is not simple, then its center is normality-large, so satisfies the property, but , being simple, doesn't. In a similar vein, if we take , then the center of is normality-large, and , which is isomorphic to the symmetric group on four elements, does not satisfy the property. (in fact, it has no cyclic normal subgroups).

## Facts

If has a subgroup that is normality-large, transitively normal, and satisfies this property (namely, every nontrivial normal subgroup of contains a cyclic normal subgroup), then also satisfies the property. This generalizes the observation that if the center is normality-large, then the group satisfies the property.