Norm (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. http://www.zentralblatt-math.org/zmath/en/search/?q=an:0009.15504&format=complete. 
• Schmidt, Roland (1994) (in en). Subgroup Lattices of Groups. Walter de Gruyter. ISBN 9783110112139. https://books.google.com/books?id=EuVadOnix5MC&q=Norm.