Baer norm

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 $$N_G(H)$$.

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.

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:

Examples in other groups
Here are some examples in non-nilpotent groups:

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.

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.