Baer norm

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 Hamiltonian 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 member of the upper central series.