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Free abelian group - Revision history
2026-07-05T08:20:04Z
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Vipul: /* Examples */
2011-01-09T21:50:20Z
<p><span dir="auto"><span class="autocomment">Examples</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:50, 9 January 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| group <math>C_0(X)</math> of singular 0-chains of a topological space <math>X</math> || underlying set of <math>X</math> || by definition plus the observation that <math>S_0(X)</math> is canonically identified with <math>X</math>.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| group <math>C_0(X)</math> of singular 0-chains of a topological space <math>X</math> || underlying set of <math>X</math> || by definition plus the observation that <math>S_0(X)</math> is canonically identified with <math>X</math>.</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>|-</div></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| [[zeroth homology group]] || [[set of path components]] || [[zeroth homology <del style="font-weight: bold; text-decoration: none;">group </del>is free on set of path components]]</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| [[zeroth homology group]] || [[set of path components]] || [[zeroth homology is free on set of path components]]</div></td></tr>
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Vipul
https://topospaces.subwiki.org/w/index.php?title=Free_abelian_group&diff=3380&oldid=prev
Vipul at 21:34, 9 January 2011
2011-01-09T21:34:17Z
<p></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:34, 9 January 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Definition==</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>===Expressive definition===</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>===Expressive definition <ins style="font-weight: bold; text-decoration: none;">with explicit generating set</ins>===</div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''.</div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>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.</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">===Definition in terms of rank===</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;"></ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">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.</ins></div></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><br></td></tr>
<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>==Examples==</div></td></tr>
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Vipul
https://topospaces.subwiki.org/w/index.php?title=Free_abelian_group&diff=3379&oldid=prev
Vipul: /* Examples */
2011-01-09T21:31:40Z
<p><span dir="auto"><span class="autocomment">Examples</span></span></p>
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<td colspan="2" style="background-color: #fff; color: #202122; text-align: center;">Revision as of 21:31, 9 January 2011</td>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Group !! Freely generating set for which it can be viewed as a free abelian group !! Proof/explanation</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>! Group !! Freely generating set for which it can be viewed as a free abelian group !! Proof/explanation</div></td></tr>
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<tr><td class="diff-marker" data-marker="−"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #ffe49c; vertical-align: top; white-space: pre-wrap;"><div>| group of [[singular chain|singular n-chain]]s in the [[singular chain complex]] || set of [[singular simplex|singular n-simplices]] || by definition</div></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div>| group <ins style="font-weight: bold; text-decoration: none;"><math>C_n(X)</math> </ins>of [[singular chain|singular n-chain]]s in the [[singular chain complex]] <ins style="font-weight: bold; text-decoration: none;">of a [[topological space]] <math>X</math> </ins>|| set <ins style="font-weight: bold; text-decoration: none;"><math>S_n(X)</math> </ins>of [[singular simplex|singular n-simplices]] || by definition</div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">|-</ins></div></td></tr>
<tr><td colspan="2" class="diff-side-deleted"></td><td class="diff-marker" data-marker="+"></td><td style="color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #a3d3ff; vertical-align: top; white-space: pre-wrap;"><div><ins style="font-weight: bold; text-decoration: none;">| group <math>C_0(X)</math> of singular 0-chains of a topological space <math>X</math> || underlying set of <math>X</math> || by definition plus the observation that <math>S_0(X)</math> is canonically identified with <math>X</math>.</ins></div></td></tr>
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<tr><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| [[zeroth homology group]] || [[set of path components]] || [[zeroth homology group is free on set of path components]]</div></td><td class="diff-marker"></td><td style="background-color: #f8f9fa; color: #202122; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #eaecf0; vertical-align: top; white-space: pre-wrap;"><div>| [[zeroth homology group]] || [[set of path components]] || [[zeroth homology group is free on set of path components]]</div></td></tr>
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Vipul
https://topospaces.subwiki.org/w/index.php?title=Free_abelian_group&diff=3378&oldid=prev
Vipul: Created page with '==Definition== ===Expressive definition=== ''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''. Suppose <math...'
2011-01-09T21:29:56Z
<p>Created page with '==Definition== ===Expressive definition=== ''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''. Suppose <math...'</p>
<p><b>New page</b></p><div>==Definition==<br />
<br />
===Expressive definition===<br />
<br />
''This is not the rigorous definition, but is the one that is most useful for explicit expressions and computations''.<br />
<br />
Suppose <math>A</math> is a set (with no additional structure necessary). The '''free abelian group''' on <math>A</math> (or the free abelian group with generating set <math>A</math>) is a group whose elements are defined as ''formal'' ''finite'' <math>\mathbb{Z}</math>-linear combinations of elements of <math>A</math>, i.e., finite sums of the form:<br />
<br />
<math>m = \sum_{a \in A} m_aa</math><br />
<br />
where <math>m_a \in \mathbb{Z}</math> for all <math>a</math> and <math>m_a \ne 0</math> for only finitely many <math>a \in A</math>. The addition rule is as follows: we group together coefficients of the same element of <math>A</math>. Thus, if <math>m = \sum_{a \in A} m_aa</math> and <math>n = \sum_{a \in A} n_aa</math>, then <math>m + n = \sum_{a \in A} (m_a + n_a)a</math>.<br />
<br />
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 <math>a + (-1)b</math> is written as <math>a - b</math>).<br />
<br />
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.<br />
<br />
==Examples==<br />
<br />
{| class="sortable" border="1"<br />
! Group !! Freely generating set for which it can be viewed as a free abelian group !! Proof/explanation<br />
|-<br />
| group of [[singular chain|singular n-chain]]s in the [[singular chain complex]] || set of [[singular simplex|singular n-simplices]] || by definition<br />
|-<br />
| [[zeroth homology group]] || [[set of path components]] || [[zeroth homology group is free on set of path components]]<br />
|}</div>
Vipul