In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, 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 Baer–Specker group, or Specker group, named after
Reinhold Baer
Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups.
Biography
Baer studied mechanical engineering f ...
and
Ernst Specker
Ernst Paul Specker (11 February 1920, Zurich – 10 December 2011, Zurich) was a Swiss mathematician. Much of his most influential work was on Quine's New Foundations, a set theory with a universal set, but he is most famous for the Kochen–Sp ...
, is an example of an infinite
Abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is commut ...
which is a building block in the structure theory of such groups.
Definition
The Baer–Specker group is the group ''B'' = Z
N of all integer sequences with componentwise addition, that is, the
direct product
In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying sets, together with a suitably defined structure on the product set. More abstractly, one ta ...
of
countably
In mathematics, a set is countable if either it is finite or it can be made in one to one correspondence with the set of natural numbers. Equivalently, a set is ''countable'' if there exists an injective function from it into the natural numbers; ...
many copies of Z. It can equivalently be described as the additive group of
formal power series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sum ...
with integer coefficients.
Properties
Reinhold Baer
Reinhold Baer (22 July 1902 – 22 October 1979) was a German mathematician, known for his work in algebra. He introduced injective modules in 1940. He is the eponym of Baer rings and Baer groups.
Biography
Baer studied mechanical engineering f ...
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.
[ attribute this result to . They write it in the form where denotes the Baer-Specker group, the star operator gives the dual group of homomorphisms to , and is the free abelian group of countable rank. They continue, "It follows that has no direct summand isomorphic to ", from which an immediate consequence is that is not free abelian.]
See also
*
Slender group
In mathematics, a slender group is a torsion-free abelian group that is "small" in a sense that is made precise in the definition below.
Definition
Let ZN denote the Baer–Specker group, that is, the group of all integer sequences, with term ...
Notes
References
*.
*.
*.
*.
*Cornelius, E. F., Jr. (2009), "Endomorphisms and product bases of the Baer-Specker group", Int'l J Math and Math Sciences, 2009, article 396475, https://www.hindawi.com/journals/ijmms/
External links
* Stefan Schröer
Baer's Result: The Infinite Product of the Integers Has No Basis
{{DEFAULTSORT:Baer-Specker Group
Abelian group theory