# Baer norm

This article is about a definition in group theory that is standard among the group theory community (or sub-community that dabbles in such things) but is not very basic or common for people outside.VIEW: Definitions built on this | Facts about this: (factscloselyrelated to Baer norm, all facts related to Baer norm) |Survey articles about this | Survey articles about definitions built on this

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This article defines a subgroup-defining function, viz., a rule that takes a group and outputs a unique subgroup

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## Definition

### Symbol-free definition

The **Baer norm** of a group is defined in the following equivalent ways:

- It is the intersection of normalizers of all its subgroups
- It is the intersection of normalizers of all cyclic subgroups.
- It is the set of those elements of the group for which the corresponding conjugation is a power automorphism.

### Definition with symbols

The **Baer norm** of a group is defined as the intersection, over all subgroups of of the groups .

### In terms of the intersect-all operator

This property is obtained by applying the intersect-all operator to the property: normalizer subgroup

View other properties obtained by applying the intersect-all operator

## Group properties

The Baer norm is a Dedekind group, i.e., it is a group in which every subgroup is normal. Conversely, every Dedekind group equals its own Baer norm.

`Further information: Baer norm is Dedekind`

## Subgroup properties

### Properties satisfied

Property | Meaning | Proof of satisfaction |
---|---|---|

Normal subgroup | ||

Hereditarily permutable subgroup | every subgroup is a permutable subgroup of the whole group | Baer norm is hereditarily permutable |

Hereditarily 2-subnormal subgroup | every subgroup is a 2-subnormal subgroup of the whole group | Baer norm is hereditarily 2-subnormal |

Characteristic subgroup | invariant under all automorphisms | Baer norm is characteristic |

Strictly characteristic subgroup | invariant under all surjective endomorphisms | Baer norm is strictly characteristic |

### Properties not satisfied

Property | Meaning | Proof of dissatisfaction |
---|---|---|

Fully invariant subgroup | invariant under all endomorphisms | Baer norm not is fully invariant |

Hereditarily normal subgroup | every subgroup is normal in the whole group | Baer norm not is hereditarily normal |

## Examples

### Dedekind groups

A Dedekind group is a group in which every subgroup is normal, or equivalently, a group that equals its own Baer norm. The finite Dedekind groups are precisely the following:

- finite abelian groups
- finite nilpotent groups whose 2-Sylow subgroup is a product of the quaternion group of order eight and an elementary abelian group, and all other Sylow subgroups are abelian.

The smallest examples of Dedekind non-abelian groups are quaternion group and direct product of Q8 and Z2.

### Examples in groups of prime power order

Here are some examples where the Baer norm is a proper subgroup:

Group part | Subgroup part | Quotient part | |
---|---|---|---|

Center of dihedral group:D8 | Dihedral group:D8 | Cyclic group:Z2 | Klein four-group |

### Examples in other groups

Here are some examples in non-nilpotent groups:

## Relation with other subgroup-defining functions

### Smaller subgroup-defining functions

Subgroup-defining function | Meaning | Proof of containment | Proof of strictness |
---|---|---|---|

Center | Elements that commute with every element | Baer norm contains center | Center not contains Baer norm |

### Larger subgroup-defining functions

Subgroup-defining function | Meaning | Proof of containment | Proof of strictness |
---|---|---|---|

Wielandt subgroup | intersection of normalizers of subnormal subgroup | Wielandt subgroup contains Baer norm | Baer norm not contains Wielandt subgroup |

Second center | second member of upper central series | Second center contains Baer norm | Baer norm not contains second center |

Centralizer of derived subgroup | centralizer of derived subgroup (commutator subgroup) | Centralizer of derived subgroup contains Baer norm | Baer norm not contains centralizer of derived subgroup |

## Related subgroup properties

- Subgroup contained in the Baer norm is a subgroup contained in the Baer norm.
- Normal subgroup contained in the Baer norm is a normal subgroup of the whole group contained in the Baer norm.

## Subgroup-defining function properties

### Reverse monotonicity

The Baer norm subgroup-defining function is weakly reverse monotone, that is, if is a subgroup of containing the Baer norm of , then the Baer norm of contains the Baer norm of .

### Idempotence and iteration

The Baer norm of a group equals its own Baer norm. A group equals its own Baer norm if and only if it is a Dedekind group, that is, every subgroup in it is normal.

### Quotient-idempotence and quotient-iteration

The quotient function corresponding to the Baer norm is *not* transitive.