Free abelian group: Difference between revisions

Revision as of 21:34, 9 January 2011

Definition

Expressive definition with explicit generating set

This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations.

Suppose $A$ is a set (with no additional structure necessary). The free abelian group on $A$ (or the free abelian group with generating set $A$ ) is a group whose elements are defined as formal finite $Z$ -linear combinations of elements of $A$ , i.e., finite sums of the form:

$m = \sum_{a \in A} m_{a} a$

where $m_{a} \in Z$ for all $a$ and $m_{a} \neq 0$ for only finitely many $a \in A$ . The addition rule is as follows: we group together coefficients of the same element of $A$ . Thus, if $m = \sum_{a \in A} m_{a} a$ and $n = \sum_{a \in A} n_{a} a$ , then $m + n = \sum_{a \in A} (m_{a} + n_{a}) a$ .

When writing a formal sum, we ignore all the terms with zero coefficient, and use subtraction to denote the addition of something with a negative coefficient (so $a + (- 1) b$ is written as $a - b$ ).

Note that if $A$ already had an additive structure, that structure is ignored for our purposes. A more precise formulation of this would be to not use the elements of $A$ themselves but a generating set equipped with a bijection to $A$ ; however, this extra layer of formality is often unnecessary.

Definition in terms of rank

Let $α$ be a cardinal (a nonnegative integer if finite, otherwise an infinite cardinal). The free abelian group of rank $α$ is defined as the free abelian group on any set of size $α$ . This is unique up to isomorphism: any bijection between two sets induces an isomorphism of the corresponding free abelian groups. In fact, something stronger is true: free abelian groups arising from sets of different cardinalities are not isomorphic, so the rank of a free abelian group is a unique cardinal.

Examples

Group	Freely generating set for which it can be viewed as a free abelian group	Proof/explanation
group $C_{n} (X)$ of singular n-chains in the singular chain complex of a topological space $X$	set $S_{n} (X)$ of singular n-simplices	by definition
group $C_{0} (X)$ of singular 0-chains of a topological space $X$	underlying set of $X$	by definition plus the observation that $S_{0} (X)$ is canonically identified with $X$ .
zeroth homology group	set of path components	zeroth homology group is free on set of path components

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 ==Definition==
-===Expressive definition===
+===Expressive definition with explicit generating set===
 ''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''.
@@ Line 14: / Line 14: @@
 Note that if <math>A</math> already had an additive structure, that structure is ''ignored'' for our purposes. A more precise formulation of this would be to not use the elements of <math>A</math> themselves but a generating set equipped with a bijection to <math>A</math>; however, this extra layer of formality is often unnecessary.
+===Definition in terms of rank===
+Let <math>\alpha</math> be a cardinal (a nonnegative integer if finite, otherwise an infinite cardinal). The free abelian group of rank <math>\alpha</math> is defined as the free abelian group on any set of size <math>\alpha</math>. This is unique up to isomorphism: any bijection between two sets induces an isomorphism of the corresponding free abelian groups. In fact, something stronger is true: free abelian groups arising from sets of different cardinalities are not isomorphic, so the ''rank'' of a free abelian group is a unique cardinal.
 ==Examples==