https://eurovision.jobogamer.xyz/index.php?action=history&feed=atom&title=Talk%3ADirect_product_of_groupsTalk:Direct product of groups - Revision history2026-06-11T04:50:54ZRevision history for this page on the wikiMediaWiki 1.45.3https://eurovision.jobogamer.xyz/index.php?title=Talk:Direct_product_of_groups&diff=3613&oldid=previmported>Ebony Jackson: /* Proposed merge of Direct sum of groups into Direct product of groups */ Renaming section to try to get the "Discuss" links to work.2026-03-07T09:42:09Z<p><span class="autocomment">Proposed merge of Direct sum of groups into Direct product of groups: </span> Renaming section to try to get the "Discuss" links to work.</p>
<p><b>New page</b></p><div>{{WikiProject banner shell|class=C|<br />
{{WikiProject Mathematics|importance = mid}}<br />
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== fibre product==<br />
--[[User:Lendormi|Lendormi]] ([[User talk:Lendormi|talk]]) 11:28, 13 January 2012 (UTC)<br />
<br />
In the section '''Fiber product''', I don't see any reason to require that the morphisms are epimorphisms.<br />
<br />
== Wikimarkup versus explicit HTML for list markup ==<br />
<br />
There's an edit war about the right way to format certain parts of the article, specifically whether wikimarkup or explicit HTML must be used. The article contains several sections where to the best of my knowledge the intended and correct indentation cannot be created using wikimarkup alone. Examples (from section, well, "Examples"):<br />
<br />
First with wikimarkup alone:<br />
<br />
<div style="margin-left: 2em"><br />
* Let {{math|'''R'''}} be the group of [[real number]]s under [[addition]]. Then the direct product {{math|'''R''' × '''R'''}} is the group of all two-component [[Euclidean vector|vectors]] {{math|(''x'', ''y'')}} under the operation of [[Euclidean vector#Addition and subtraction|vector addition]]:<br />
::{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) &nbsp;{{=}}&nbsp; (''x''<sub>1</sub> + ''x''<sub>2</sub>, ''y''<sub>1</sub> + ''y''<sub>2</sub>)}}.<br />
* Let {{math|''G''}} and {{math|''H''}} be [[cyclic group]]s with two elements each:<br />
<br />
:{|<br />
|-<br />
|<br />
<!-- G nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''G''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''a''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''a''}}<br />
|-<br />
! {{math|''a''}}<br />
| {{math|''a''}} || {{math|1}}<br />
|}<br />
<!-- / G nested table --><br />
| style="width:15px;" |<br />
|<br />
<!-- H nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''H''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''b''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''b''}}<br />
|-<br />
! {{math|''b''}}<br />
| {{math|''b''}} || {{math|1}}<br />
|}<br />
<!-- / H nested table --><br />
|}<br />
<br />
Then the direct product {{math|''G'' × ''H''}} is [[group isomorphism|isomorphic]] to the [[Klein four-group]]:<br />
<br />
{| class="wikitable" style="text-align:center; width:250px; height:175px;"<br />
|+ {{math|''G'' × ''H''}}<br />
|-<br />
! ∙<br />
! {{math|(1, 1)}} !! {{math|(''a'', 1)}} !! {{math|(1, ''b'')}} !! {{math|(''a'', ''b'')}}<br />
|-<br />
! {{math|(1, 1)}}<br />
| {{math|(1, 1)}} || {{math|(''a'', 1)}} || {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}}<br />
|-<br />
! {{math|(''a'', 1)}}<br />
| {{math|(''a'', 1)}} || {{math|(1, 1)}} || {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}}<br />
|-<br />
! {{math|(1, ''b'')}}<br />
| {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}} || {{math|(1, 1)}} || {{math|(''a'', 1)}}<br />
|-<br />
! {{math|(''a'', ''b'')}}<br />
| {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}} || {{math|(''a'', 1)}} || {{math|(1, 1)}}<br />
|}<br />
</div><br />
<br />
This is supposed to be a two-element unordered list. However, the second list entry is broken semantically and visually as the tables are not interpreted as part of the list entry, and their indentation is wrong. Now using HTML:<br />
<br />
<div style="margin-left: 2em"><br />
<ul><br />
<li>Let {{math|'''R'''}} be the group of [[real number]]s under [[addition]]. Then the direct product {{math|'''R''' × '''R'''}} is the group of all two-component [[Euclidean vector|vectors]] {{math|(''x'', ''y'')}} under the operation of [[Euclidean vector#Addition and subtraction|vector addition]]:<br />
:{{math|(''x''<sub>1</sub>, ''y''<sub>1</sub>) + (''x''<sub>2</sub>, ''y''<sub>2</sub>) &nbsp;{{=}}&nbsp; (''x''<sub>1</sub> + ''x''<sub>2</sub>, ''y''<sub>1</sub> + ''y''<sub>2</sub>)}}.<br />
</li><br />
<li>Let {{math|''G''}} and {{math|''H''}} be [[cyclic group]]s with two elements each:<br />
<br />
{|<br />
|-<br />
|<br />
<!-- G nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''G''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''a''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''a''}}<br />
|-<br />
! {{math|''a''}}<br />
| {{math|''a''}} || {{math|1}}<br />
|}<br />
<!-- / G nested table --><br />
| style="width:15px;" |<br />
|<br />
<!-- H nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''H''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''b''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''b''}}<br />
|-<br />
! {{math|''b''}}<br />
| {{math|''b''}} || {{math|1}}<br />
|}<br />
<!-- / H nested table --><br />
|}<br />
<br />
Then the direct product {{math|''G'' × ''H''}} is [[group isomorphism|isomorphic]] to the [[Klein four-group]]:<br />
<br />
{| class="wikitable" style="text-align:center; width:250px; height:175px;"<br />
|+ {{math|''G'' × ''H''}}<br />
|-<br />
! ∙<br />
! {{math|(1, 1)}} !! {{math|(''a'', 1)}} !! {{math|(1, ''b'')}} !! {{math|(''a'', ''b'')}}<br />
|-<br />
! {{math|(1, 1)}}<br />
| {{math|(1, 1)}} || {{math|(''a'', 1)}} || {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}}<br />
|-<br />
! {{math|(''a'', 1)}}<br />
| {{math|(''a'', 1)}} || {{math|(1, 1)}} || {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}}<br />
|-<br />
! {{math|(1, ''b'')}}<br />
| {{math|(1, ''b'')}} || {{math|(''a'', ''b'')}} || {{math|(1, 1)}} || {{math|(''a'', 1)}}<br />
|-<br />
! {{math|(''a'', ''b'')}}<br />
| {{math|(''a'', ''b'')}} || {{math|(1, ''b'')}} || {{math|(''a'', 1)}} || {{math|(1, 1)}}<br />
|}<br />
</li></ul><br />
</div><br />
<br />
Here the indentation of the tables is correct, and the second list entry is not broken. I don't see how this can be accomplished using wikimarkup alone, and [[Help:List]] appears to agree, as far as I can see. – [[User:Tea2min|Tea2min]] ([[User talk:Tea2min|talk]]) 11:22, 1 June 2016 (UTC)<br />
<br />
:How about this -- no div-style, and just plain wiki markup:<br />
:* Let {{math|''G''}} and {{math|''H''}} be [[cyclic group]]s with two elements each:<br />
<br />
::{|<br />
|-<br />
|<br />
<!-- G nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''G''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''a''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''a''}}<br />
|-<br />
! {{math|''a''}}<br />
| {{math|''a''}} || {{math|1}}<br />
|}<br />
<!-- / G nested table --><br />
| style="width:15px;" |<br />
|<br />
<!-- H nested table --><br />
{| class="wikitable" style="text-align:center; width:90px; height:90px;"<br />
|+ {{math|''H''}}<br />
|-<br />
! ∙<br />
! {{math|1}} !! {{math|''b''}}<br />
|-<br />
! {{math|1}}<br />
| {{math|1}} || {{math|''b''}}<br />
|-<br />
! {{math|''b''}}<br />
| {{math|''b''}} || {{math|1}}<br />
|}<br />
<!-- / H nested table --><br />
|}<br />
<br />
Then the direct product {{math|''G'' × ''H''}} is [[group isomorphism|isomorphic]] to the [[Klein four-group]]:<br />
<br />
And so on. Its wikimarkup, but I just put a single colon in front of the table to indent it. No raw html markup. 14:29, 3 August 2016 (UTC)<br />
<br />
</li></ul></dd></dl><br />
</li></ul></dd></dl><br />
</li></ul></dd></dl><br />
<br />
== Proposed merge ==<br />
<br />
The proposal is to merge [[Direct sum of groups]] into [[Direct product of groups]].<br />
<br />
The article [[Direct sum of groups]] is describing the same construction as [[Direct product of groups]]. There is a difference in the case of infinite direct sums, but in the literature those are almost always in the setting of infinite direct sums of ''abelian'' groups (or vector spaces or modules), and those are amply described already at articles such as [[Direct sum]] and [[Direct sum of modules]]. See [[WP:CONTENTFORK]]. —[[User:Ebony Jackson|Ebony Jackson]] ([[User talk:Ebony Jackson|talk]]) 20:36, 27 February 2026 (UTC)</div>imported>Ebony Jackson