Norm (group)

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. Der Kern, eine charakteristische Untergruppe, Compositio Math. 1, 254-283. Zbl9.15504
  • Schmidt, Roland. Subgroup Lattices of Groups. de Gruyter, 1994
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