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

Property theory

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.

Relation with other subgroup-defining functions

The Baer norm of a group contains the center of the group. Conjugation by any element in the Baer norm is a power automorphism, hence it commutes with all inner automorphisms. Thus, the Baer norm is contained in the second center, i.e., the second member of the upper central series.

For full proof, refer: Baer norm contains center, second center contains Baer norm