mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

, in the field of

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ( ...

, the norm of a group is the intersection of the

normalizer In mathematics, especially group theory, the centralizer (also called commutant) of a subset ''S'' in a group ''G'' is the set \operatorname_G(S) of elements of ''G'' that commute with every element of ''S'', or equivalently, the set of ele ...

s of all its

subgroup In group theory, a branch of mathematics, a subset of a group G is a subgroup of G if the members of that subset form a group with respect to the group operation in G. Formally, given a group (mathematics), group under a binary operation ...

s. 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

* * Group theory Functional subgroups {{group-theory-stub