Baer norm

Definition

Symbol-free definition

The Baer norm of a group is the intersection of normalizers of all its subgroups. Equivalently, it is the intersection of normalizers of all cyclic subgroups.

The Baer norm can also be characterized as 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 $N_{G}(H)$.

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.