Norm (group)
Encyclopedia
In mathematics
, in the field of group theory
, the norm of a group
is the intersection of the normalizers of all its subgroup
s. This is also termed the Baer norm, after Reinhold Baer
.
The following facts are true for the Baer norm:
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, in the field of group theory
Group theory
In mathematics and abstract algebra, group theory studies the algebraic structures known as groups.The concept of a group is central to abstract algebra: other well-known algebraic structures, such as rings, fields, and vector spaces can all be seen as groups endowed with additional operations and...
, the norm of a group
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
is the intersection of the normalizers of all its subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H x H is a group operation on H...
s. This is also termed the Baer norm, after Reinhold Baer
Reinhold Baer
Reinhold Baer was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings....
.
The following facts are true for the Baer norm:
- It is a characteristic subgroupCharacteristic subgroupIn mathematics, particularly in the area of abstract algebra known as group theory, a characteristic subgroup is a subgroup that is invariant under all automorphisms of the parent group. Because conjugation is an automorphism, every characteristic subgroup is normal, though not every normal...
. - It contains the centerCenter (group theory)In abstract algebra, the center of a group G, denoted Z,The notation Z is from German Zentrum, meaning "center". is the set of elements that commute with every element of G. In set-builder notation,...
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 groupQuaternion groupIn group theory, the quaternion group is a non-abelian group of order eight, isomorphic to a certain eight-element subset of the quaternions under multiplication...
. - If it contains an element of infinite order, then it is equal to the center of the group.