https://dictionary.iucr.org/index.php?title=Semidirect_product&feed=atom&action=history
Semidirect product - Revision history
2024-03-28T11:03:04Z
Revision history for this page on the wiki
MediaWiki 1.30.0
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=4700&oldid=prev
BrianMcMahon: Tidied translations.
2017-12-15T11:32:16Z
<p>Tidied translations.</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 11:32, 15 December 2017</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; 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><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="<del class="diffchange diffchange-inline">brown</del>"><del class="diffchange diffchange-inline">Полупрямое произведение</del></font> (''<del class="diffchange diffchange-inline">Ru</del>''). <font color="<del class="diffchange diffchange-inline">black</del>"><del class="diffchange diffchange-inline">Prodotto semidiretto</del></font> (''<del class="diffchange diffchange-inline">It</del>''). <font color="<del class="diffchange diffchange-inline">purple</del>"><del class="diffchange diffchange-inline">準直積</del></font> (''<del class="diffchange diffchange-inline">Ja</del>''). <font color="green">Producto semidirecto</font> (''Sp'').</div></td><td class='diff-marker'>+</td><td style="color:black; 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><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="<ins class="diffchange diffchange-inline">black</ins>"><ins class="diffchange diffchange-inline">Prodotto semidiretto</ins></font> (''<ins class="diffchange diffchange-inline">It</ins>''). <font color="<ins class="diffchange diffchange-inline">purple</ins>"><ins class="diffchange diffchange-inline">準直積</ins></font> (''<ins class="diffchange diffchange-inline">Ja</ins>''). <font color="<ins class="diffchange diffchange-inline">brown</ins>"><ins class="diffchange diffchange-inline">Полупрямое произведение</ins></font> (''<ins class="diffchange diffchange-inline">Ru</ins>''). <font color="green">Producto semidirecto</font> (''Sp'').</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=4595&oldid=prev
BrianMcMahon: Tidied translations.
2017-11-20T08:42:21Z
<p>Tidied translations.</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 08:42, 20 November 2017</td>
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<tr><td class='diff-marker'>−</td><td style="color:black; 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><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="<del class="diffchange diffchange-inline">green</del>"><del class="diffchange diffchange-inline">Producto semidirecto</del></font> (''<del class="diffchange diffchange-inline">Sp</del>''). <font color="<del class="diffchange diffchange-inline">brown</del>"><del class="diffchange diffchange-inline">Полупрямое произведение</del></font> (''<del class="diffchange diffchange-inline">Ru</del>''). <font color="<del class="diffchange diffchange-inline">black</del>"><del class="diffchange diffchange-inline">Prodotto semidiretto</del></font> (''<del class="diffchange diffchange-inline">It</del>''). <font color="<del class="diffchange diffchange-inline">purple</del>"><del class="diffchange diffchange-inline">準直積</del></font> (''<del class="diffchange diffchange-inline">Ja</del>'').  </div></td><td class='diff-marker'>+</td><td style="color:black; 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><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="<ins class="diffchange diffchange-inline">brown</ins>"><ins class="diffchange diffchange-inline">Полупрямое произведение</ins></font> (''<ins class="diffchange diffchange-inline">Ru</ins>''). <font color="<ins class="diffchange diffchange-inline">black</ins>"><ins class="diffchange diffchange-inline">Prodotto semidiretto</ins></font> (''<ins class="diffchange diffchange-inline">It</ins>''). <font color="<ins class="diffchange diffchange-inline">purple</ins>"><ins class="diffchange diffchange-inline">準直積</ins></font> (''<ins class="diffchange diffchange-inline">Ja</ins>''). <font color="<ins class="diffchange diffchange-inline">green</ins>"><ins class="diffchange diffchange-inline">Producto semidirecto</ins></font> (''<ins class="diffchange diffchange-inline">Sp</ins>'').</div></td></tr>
<tr><td colspan="2"> </td><td class='diff-marker'>+</td><td style="color:black; 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> </div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=4103&oldid=prev
BrianMcMahon: Style edits to align with printed edition
2017-05-17T10:32:17Z
<p>Style edits to align with printed edition</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 10:32, 17 May 2017</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l1" >Line 1:</td>
<td colspan="2" class="diff-lineno">Line 1:</td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="green">Producto semidirecto</font> (''Sp''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja'').  </div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="green">Producto semidirecto</font> (''Sp''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja'').  </div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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><del style="font-weight: bold; text-decoration: none;"></del></div></td><td colspan="2"> </td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e.<del class="diffchange diffchange-inline">, </del>''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to <del class="diffchange diffchange-inline">say </del>that:</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (<ins class="diffchange diffchange-inline">''</ins>i.e.<ins class="diffchange diffchange-inline">'' </ins>''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[Group homomorphism|kernel]] is ''N''. This is equivalent to <ins class="diffchange diffchange-inline">saying </ins>that:</div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where <del class="diffchange diffchange-inline">"</del>1<del class="diffchange diffchange-inline">" </del>is identity element of ''G''<del class="diffchange diffchange-inline"> </del>)</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where <ins class="diffchange diffchange-inline">'</ins>1<ins class="diffchange diffchange-inline">' </ins>is <ins class="diffchange diffchange-inline">the </ins>identity element of ''G'')<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''</div></td><td class='diff-marker'>+</td><td style="color:black; 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>* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''<ins class="diffchange diffchange-inline">.</ins></div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>One also says that <del class="diffchange diffchange-inline">"</del>''G'' ''splits'' over ''N''<del class="diffchange diffchange-inline">"</del>.</div></td><td class='diff-marker'>+</td><td style="color:black; 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>One also says that <ins class="diffchange diffchange-inline">`</ins>''G'' ''splits'' over ''N''<ins class="diffchange diffchange-inline">'</ins>.</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>[[Category:Fundamental crystallography]]</div></td></tr>
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BrianMcMahon
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=2830&oldid=prev
MassimoNespolo: broken link
2008-12-21T18:31:25Z
<p>broken link</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 18:31, 21 December 2008</td>
</tr><tr><td colspan="2" class="diff-lineno" id="mw-diff-left-l4" >Line 4:</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[<del class="diffchange diffchange-inline">kernel (algebra)</del>|kernel]] is ''N''. This is equivalent to say that:</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[group homomorphism|homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[<ins class="diffchange diffchange-inline">Group homomorphism</ins>|kernel]] is ''N''. This is equivalent to say that:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}</div></td></tr>
</table>
MassimoNespolo
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=2620&oldid=prev
MassimoNespolo: link
2007-05-29T10:00:00Z
<p>link</p>
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<td colspan="2" style="background-color: white; color:black; text-align: center;">← Older revision</td>
<td colspan="2" style="background-color: white; color:black; text-align: center;">Revision as of 10:00, 29 May 2007</td>
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<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"></td></tr>
<tr><td class='diff-marker'>−</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:</div></td><td class='diff-marker'>+</td><td style="color:black; 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>Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[<ins class="diffchange diffchange-inline">group homomorphism|</ins>homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )</div></td></tr>
<tr><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}</div></td><td class='diff-marker'> </td><td style="background-color: #f9f9f9; color: #333333; font-size: 88%; border-style: solid; border-width: 1px 1px 1px 4px; border-radius: 0.33em; border-color: #e6e6e6; vertical-align: top; white-space: pre-wrap;"><div>* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}</div></td></tr>
</table>
MassimoNespolo
https://dictionary.iucr.org/index.php?title=Semidirect_product&diff=2618&oldid=prev
MassimoNespolo at 09:40, 29 May 2007
2007-05-29T09:40:28Z
<p></p>
<p><b>New page</b></p><div><font color="blue">Produit semi-direct</font> (''Fr''). <font color="red">Semidirektes Produkt</font> (''Ge''). <font color="green">Producto semidirecto</font> (''Sp''). <font color="brown">Полупрямое произведение</font> (''Ru''). <font color="black">Prodotto semidiretto</font> (''It''). <font color="purple">準直積</font> (''Ja''). <br />
<br />
<br />
In group theory, a '''semidirect product''' describes a particular way in which a group can be put together from two subgroups, one of which is [[normal subgroup|normal]].<br />
<br />
Let ''G'' be a group, ''N'' a [[normal subgroup]] of ''G'' (i.e., ''N'' &#x25C1; ''G'') and ''H'' a [[subgroup]] of ''G''. ''G'' is a '''semidirect product''' of ''N'' and ''H'' if there exists a [[homomorphism]] ''G'' &rarr; ''H'' which is the identity on ''H'' and whose [[kernel (algebra)|kernel]] is ''N''. This is equivalent to say that:<br />
* ''G'' = ''NH'' and ''N'' &cap; ''H'' = {1} (where "1" is identity element of ''G'' )<br />
* ''G'' = ''HN'' and ''N'' &cap; ''H'' = {1}<br />
* Every element of ''G'' can be written as a unique product of an element of ''N'' and an element of ''H''<br />
* Every element of ''G'' can be written as a unique product of an element of ''H'' and an element of ''N''<br />
<br />
One also says that "''G'' ''splits'' over ''N''".<br />
<br />
[[Category:Fundamental crystallography]]</div>
MassimoNespolo