definition of free Abelian group
Topics
About: group
The table of contents of this article
Starting Context
- The reader knows a definition of Abelian group.
- The reader knows a definition of subgroup generated by subset of group.
Target Context
- The reader will have a definition of free Abelian group.
Orientation
There is a list of definitions discussed so far in this site.
There is a list of propositions discussed so far in this site.
Main Body
1: Structured Description
Here is the rules of Structured Description.
Entities:
\(*G\): \(\in \{\text{ the Abelian groups }\}\), expressed as additive
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Conditions:
\(\exists B \subseteq G (\forall \{b_1, ..., b_n\} \subseteq B (z^1 b_1 + ... + z^n b_n = 0 \implies z^j = 0) \land (B) = G)\)
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\(z^j b_j\) means \(b_j + ... + b_j\) with \(z^j\) \(b_j\) s or \((- b_j) + ... + (- b_j)\) with \(- z^j\) \(b_j\) s, so, \(z^j \in \mathbb{Z}\).
\(B\) is called "basis of \(G\)".
2: Note
"expressed as additive" is just a declaration for our notion, which does not really restrict \(G\) in any way: it is a matter of expressing the group operation as \(+\).
As has been seen in the definition of subgroup generated by subset of group, \(G = (B)\) is the set of all the finite multiplications (which are additions in additive Abelian group) of the elements of \(B\) and their inverses, so, each element of \(G\) is expressed as \(z^1 b_1 + ... + z^n b_n\), which is a unique expression, because when \(z^1 b_1 + ... + z^n b_n = z'^1 b_1 + ... + z'^n b_n\), \((z^1 - z'^1) b_1 + ... + (z^n - z'^n) b_n = 0\), which implies that \(z^j = z'^j\).
