# Group in which every permutable subgroup is normal

From Groupprops

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 permutable subgroup is normal** is a group such that every permutable subgroup of the group is a normal subgroup of the group.

## Formalisms

### In terms of the subgroup property collapse operator

This group property can be defined in terms of the collapse of two subgroup properties. In other words, a group satisfies this group property if and only if every subgroup of it satisfying the first property (permutable subgroup) satisfies the second property (normal subgroup), and vice versa.

View other group properties obtained in this way

## Relation with other properties

### Stronger properties

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

Abelian group | any two elements commute | |FULL LIST, MORE INFO | ||

Dedekind group | every subgroup is a normal subgroup | |FULL LIST, MORE INFO | ||

Finite T-group | |FULL LIST, MORE INFO |