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| − | A [[Simple finite group|simple finite group]] that does not belong to any of the known infinite series of simple finite groups. The twenty-six sporadic simple groups are listed in the following table.''''''<table border="0" cellpadding="0" cellspacing="0" style="background-color:black;"> <tr><td> <table border="0" cellspacing="1" cellpadding="4" style="background-color:black;"> <tbody> <tr> <td colname="1" style="background-color:white;" colspan="1">notation</td> <td colname="2" style="background-color:white;" colspan="1">name</td> <td colname="3" style="background-color:white;" colspan="1">order</td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868701.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868702.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868703.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868704.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868705.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Mathieu groups</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868706.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868707.png" /></td> <td colname="2" style="background-color:white;"
| + | Post $\TeX$ remarks |
| − | colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868708.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s0868709.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687010.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687011.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Janko group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687012.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687014.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Hall–Janko group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687015.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687017.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Higman–Janko–McKay group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687018.png" /></td> </tr> <tr> <td colname="1" style="background-color:white
| + | * I could not edit this page using my usual method (wikipediafs) so I could not fix the refs, could someone do it for me? |
| − | ;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687019.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Janko group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687020.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687021.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687022.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687023.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687025.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Conway groups</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687026.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687027.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687028.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687029.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687030.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/
| + | * It might be nice to have some extra columns here |
| − | s086870/s08687031.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687032.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687033.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687034.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Fischer groups</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687035.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687037.png" /></td> <td colname="2" style="background-color:white;" colspan="1"></td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687038.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687039.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Higman–Sims group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687040.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687041.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687042.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Held–Higman–McKay group</td> <td colname="3" style="background-c
| + | ** reference to the relevant paper |
| − | olor:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687043.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687044.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Suzuki group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687045.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687046.png" /></td> <td colname="2" style="background-color:white;" colspan="1">McLaughlin group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687047.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687048.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Lyons group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687049.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687050.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Rudvalis group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687051.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687052.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687053.png" /></td> <td colname="2" style="backgrou
| + | ** date of discovery |
| − | nd-color:white;" colspan="1">O'Nan–Sims group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687054.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687056.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Monster, Fischer–Griess group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687057.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687059.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Baby monster</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687060.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687062.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687063.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Thompson group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687064.png" /></td> </tr> <tr> <td colname="1" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687065.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s08
| + | ** order in decimal |
| − | 6870/s08687066.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687067.png" /></td> <td colname="2" style="background-color:white;" colspan="1">Harada–Norton group</td> <td colname="3" style="background-color:white;" colspan="1"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687068.png" /></td> </tr> </tbody> </table>
| + | * possibly allow sorting by some of these columns? |
| | + | I am seriously out of my depth on this topic, so I will leave it to others to consider whether these are worth doing ... -- [[User:Jjg|Jjg]] 21:12, 30 April 2012 (CEST) |
| | | | |
| − | </td></tr> </table>
| + | : This math is out of my competence, but anyway I wonder, why wikipediafs failed here? --[[User:Boris Tsirelson|Boris Tsirelson]] 21:34, 30 April 2012 (CEST) |
| | | | |
| − | ====References====
| + | :: Not sure, it may be a problem at my end, but now I cannot edit any pages via wikipediafs (even ones that I was editing yesterday). Very strange. --[[User:Jjg|Jjg]] 23:29, 30 April 2012 (CEST) |
| − | <table><TR><TD valign="top">[1]</TD> <TD valign="top"> S.A. Syskin, "Abstract properties of the simple sporadic groups" ''Russian Math. Surveys'' , '''35''' : 5 (1980) pp. 209–246 ''Uspekhi Mat. Nauk'' , '''35''' : 5 (1980) pp. 181–212</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top"> M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980)</TD></TR></table>
| |
| | | | |
| | + | ::: I don't know anything about wikipediafs, but its [http://en.wikipedia.or /wiki/WikipediaFS#Disadvantages wp page] says its version 0.3 does not work with current wp. I prefer working with browser extensions like "Edit with emacs" (chromium) and/or "It's all text" (firefox). --[[User:Ulf Rehmann|Ulf Rehmann]] 09:58, 1 May 2012 (CEST) |
| | | | |
| | + | :::: The most recent version works fine, it has the disadvantage that it must login to make each each "commit", and does not give a lot of feedback when it fails (the above failure was due to the recent login problems), but at least it does work -- I spent a full evening trying to coerce "It's all text" to play nice, and I really don't want to change browser just to edit a page... --[[User:Jjg|Jjg]] 19:24, 2 May 2012 (CEST) |
| | | | |
| − | ====Comments====
| + | A lot of things could be added here. Each of the sporadic group names should have its own page. We only have the Mathieu groups as a distinguished page. --[[User:Ulf Rehmann|Ulf Rehmann]] 01:02, 1 May 2012 (CEST) |
| − | The recent classification of the finite simple groups (1981) has led to the conclusion that — up to a uniqueness proof for the Monster as the only simple group of its order with certain additional properties — every non-Abelian finite simple group is isomorphic to: an [[Alternating group|alternating group]] on at least <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687069.png" /> letters, a group of (twisted or untwisted) Lie type, or one of the above <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s086/s086870/s08687070.png" /> sporadic groups. See [[#References|[a2]]] for a discussion of the proof.
| |
| | | | |
| − | ====References====
| + | I like to propose that the word "group" be dropped from the second column and put in the header instead; eg |
| − | <table><TR><TD valign="top">[a1]</TD> <TD valign="top"> J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top"> D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982)</TD></TR></table>
| + | $\newcommand{\d}{\cdot}$ |
| | + | {| class="wikitable" style="margin: 1em auto 1em auto;" |
| | + | |+ The twenty-six sporadic simple groups |
| | + | ! Notation |
| | + | ! Name (group) |
| | + | ! Order |
| | + | |- |
| | + | | $J_1$ |
| | + | | Janko |
| | + | | $2^3\d 3\d 5\d 7\d 11\d 19$ |
| | + | |- |
| | + | |} |
| | + | This is normal typographic practice in tables for units, for example. It would have the advantage that it would free up some horizontal space for other columns, as suggested above. --[[User:Jjg|Jjg]] 19:24, 2 May 2012 (CEST) |
Post $\TeX$ remarks
- I could not edit this page using my usual method (wikipediafs) so I could not fix the refs, could someone do it for me?
- It might be nice to have some extra columns here
- reference to the relevant paper
- date of discovery
- order in decimal
- possibly allow sorting by some of these columns?
I am seriously out of my depth on this topic, so I will leave it to others to consider whether these are worth doing ... -- Jjg 21:12, 30 April 2012 (CEST)
- This math is out of my competence, but anyway I wonder, why wikipediafs failed here? --Boris Tsirelson 21:34, 30 April 2012 (CEST)
- Not sure, it may be a problem at my end, but now I cannot edit any pages via wikipediafs (even ones that I was editing yesterday). Very strange. --Jjg 23:29, 30 April 2012 (CEST)
- I don't know anything about wikipediafs, but its /wiki/WikipediaFS#Disadvantages wp page says its version 0.3 does not work with current wp. I prefer working with browser extensions like "Edit with emacs" (chromium) and/or "It's all text" (firefox). --Ulf Rehmann 09:58, 1 May 2012 (CEST)
- The most recent version works fine, it has the disadvantage that it must login to make each each "commit", and does not give a lot of feedback when it fails (the above failure was due to the recent login problems), but at least it does work -- I spent a full evening trying to coerce "It's all text" to play nice, and I really don't want to change browser just to edit a page... --Jjg 19:24, 2 May 2012 (CEST)
A lot of things could be added here. Each of the sporadic group names should have its own page. We only have the Mathieu groups as a distinguished page. --Ulf Rehmann 01:02, 1 May 2012 (CEST)
I like to propose that the word "group" be dropped from the second column and put in the header instead; eg
$\newcommand{\d}{\cdot}$
The twenty-six sporadic simple groups
| Notation
|
Name (group)
|
Order
|
| $J_1$
|
Janko
|
$2^3\d 3\d 5\d 7\d 11\d 19$
|
This is normal typographic practice in tables for units, for example. It would have the advantage that it would free up some horizontal space for other columns, as suggested above. --Jjg 19:24, 2 May 2012 (CEST)