# Baer norm is hereditarily permutable

This article gives the statement, and possibly proof, of the fact that for any group, the subgroup obtained by applying a given subgroup-defining function (i.e., Baer norm) always satisfies a particular subgroup property (i.e., hereditarily permutable subgroup)}
View subgroup property satisfactions for subgroup-defining functions  View subgroup property dissatisfactions for subgroup-defining functions

## Statement

### Verbal statement

The Baer norm of any group is a hereditarily permutable subgroup: every subgroup of it is permutable in the whole group.

### Statement with symbols

Suppose  is a group and  is the Baer norm of :  is the intersection of Normalizer (?)s of all the subgroups of . Then,  is a hereditarily permutable subgroup. In other words, if  is a subgroup of , and  is a subgroup of , then  and  permute: .

## Definitions used

### Baer norm

Further information: Baer norm

The Baer norm of a group is defined as the intersection of normalizers of all its subgroups. In symbols, for a group , the Baer norm  is given by:

.

### Hereditarily permutable subgroup

Further information: Hereditarily permutable subgroup

A subgroup  is termed hereditarily permutable if for every subgroup  and every subgroup , , i.e.,  and  are permuting subgroups.

## Related facts

• Baer norm not is hereditarily normal: The Baer norm of a group need not be hereditarily normal: it may have subgroups that are not normal in the whole group.
• Baer norm is Dedekind: Every subgroup of the Baer norm is normal inside the Baer norm (though, as pointed above, it need not be normal in the whole group).

## Proof

Given: A group  with Baer norm . A subgroup  and a subgroup .

To prove: .

Proof:

1. : Since  is the intersection of normalizers of all subgroups of , . In particular, since , we have .
2. : Since , we have  for all . Thus, , which is defined as the union of the , must equal , which is defined as the union of the .