# Group in which any two normal subgroups are comparable

From Groupprops

## Contents

## Definition

A group is said to be a **group in which any two normal subgroups are comparable** if any two normal subgroups of the group can be compared, or in other words, if its lattice of normal subgroups is a totally ordered set. In other words, given any two normal subgroups of the group, one of them must lie completely inside the other.

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

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Simple group | no proper nontrivial normal subgroup | |||

Cyclic group of prime power order | cyclic group, order is a prime power |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

group in which any two characteristic subgroups are comparable | any two characteristic subgroups are comparable | |||

group in which every normal subgroup is characteristic (at least for a finite group) | ||||

group whose center is comparable with all normal subgroups |