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ehrenfest
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Homework Statement
What is the point of giving free abelian groups a special name if they are all isomorphic to Z times Z times Z ... times Z for r factors of Z, where r is the rank of the basis?
A free Abelian group is a mathematical structure that consists of a set of elements and an operation, typically addition, that satisfies certain properties. These properties include closure, associativity, commutativity, identity, and inverse. The "free" part means that the elements in the group can be freely combined to form new elements and the group does not have any additional constraints or relations.
When a free Abelian group is isomorphic to Z x Z...xZ, it means that the group is equivalent to the direct product of an infinite number of copies of the integers. This means that the group has the same structure and properties as the direct product of infinite copies of the integers, even though the elements and operations may be different.
The easiest way to determine if a free Abelian group is isomorphic to Z x Z...xZ is by using the fundamental theorem of finitely generated Abelian groups. This theorem states that any finitely generated Abelian group can be written as the direct product of a finite number of cyclic groups. If the free Abelian group has an infinite number of generators, then it can be written as an infinite direct product of cyclic groups, which is isomorphic to Z x Z...xZ.
Yes, there are other groups that are isomorphic to Z x Z...xZ. For example, the direct product of any infinite number of copies of a finite cyclic group is isomorphic to Z x Z...xZ. Additionally, the direct product of any infinite number of copies of a countable group is also isomorphic to Z x Z...xZ.
The isomorphism between free Abelian groups and Z x Z...xZ is important because it allows us to understand and study free Abelian groups by using our knowledge of the integers. This is especially useful in applications to algebraic topology and algebraic geometry, where free Abelian groups are fundamental objects. The isomorphism also helps to simplify calculations and proofs involving free Abelian groups.