# Hereditarily permutable subgroup

From Groupprops

BEWARE!This term is nonstandard and is being used locally within the wiki. [SHOW MORE]

This article defines a subgroup property: a property that can be evaluated to true/false given a group and a subgroup thereof, invariant under subgroup equivalence. View a complete list of subgroup properties[SHOW MORE]

## Definition

### Symbol-free definition

A subgroup of a group is termed **hereditarily permutable** if every subgroup of that subgroup is a permutable subgroup in the whole group.

### Definition with symbols

A subgroup of a group is termed **hereditarily permutable** in if whenever , is a permutable subgroup of .

## Formalisms

### In terms of the hereditarily operator

This property is obtained by applying the hereditarily operator to the property: permutable subgroup

View other properties obtained by applying the hereditarily operator

## Relation with other properties

### Stronger properties

### Weaker properties

- Right-transitively permutable subgroup
- Intersection-transitively permutable subgroup
- Permutable subgroup

## Facts

- The Baer norm of a group, defined as the intersection of normalizers of all its subgroups, is hereditarily permutable.
`For full proof, refer: Baer norm is hereditarily permutable`