Baer–Specker group
Encyclopedia
In mathematics
, in the field of group theory
, the Baer–Specker group, or Specker group, is an example of an infinite Abelian group
which is a building block in the structure theory of such groups.
of countably many copies of Z.
proved in 1937 that this group is not free abelian
; Specker proved in 1950 that every countable subgroup of B is free abelian.
The group of homomorphisms from the Baer–Specker group to a free abelian group of finite rank is a free abelian group of countable rank. This provides another proof that the group is not free.
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 Baer–Specker group, or Specker group, is an example of an infinite Abelian group
Abelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
which is a building block in the structure theory of such groups.
Definition
The Baer–Specker group is the group B = ZN of all integer sequences with componentwise addition, that is, the direct productDirect product of groups
In the mathematical field of group theory, the direct product is an operation that takes two groups and and constructs a new group, usually denoted...
of countably many copies of Z.
Properties
Reinhold BaerReinhold 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....
proved in 1937 that this group is not free abelian
Free abelian group
In abstract algebra, a free abelian group is an abelian group that has a "basis" in the sense that every element of the group can be written in one and only one way as a finite linear combination of elements of the basis, with integer coefficients. Hence, free abelian groups over a basis B are...
; Specker proved in 1950 that every countable subgroup of B is free abelian.
The group of homomorphisms from the Baer–Specker group to a free abelian group of finite rank is a free abelian group of countable rank. This provides another proof that the group is not free.
External links
- Stefan Schröer, Baer's Result: The Infinite Product of the Integers Has No Basis