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.
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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 $G$ is defined as the intersection, over all subgroups $H$ of $G$ of the groups $N_{G}(H)$.

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 $K$ is a subgroup of $G$ containing the Baer norm of $G$, then the Baer norm of $K$ contains the Baer norm of $H$.

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.