Norm (group)

For the mapping to real numbers, see Norm (abelian group). For the Swiss typography team, see Norm (graphic design group).

In mathematics, in the field of group theory, the norm of a group is the intersection of the normalizers of all its subgroups. This is also termed the Baer norm, after Reinhold Baer.

The following facts are true for the Baer norm:

• It is a characteristic subgroup.
• It contains the center of the group.
• It is contained inside the second term of the upper central series.
• It is a Dedekind group, so is either abelian or has a direct factor isomorphic to the quaternion group.
• If it contains an element of infinite order, then it is equal to the center of the group.

References

• Baer, Reinhold (1934). "Der Kern, eine charakteristische Untergruppe". Compositio Mathematica. 1: 254–283.
• Schmidt, Roland (1994). Subgroup Lattices of Groups. Walter de Gruyter. ISBN 9783110112139.