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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;" 
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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.
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
 
;" 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/
 
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
 
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
 
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
 
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>
 
 
 
</td></tr> </table>
 
  
 
====References====
 
====References====

Revision as of 16:35, 30 April 2012

A 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.

References

[1] 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
[2] M. Aschbacher, "The finite simple groups and their classification" , Yale Univ. Press (1980)


Comments

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 on at least letters, a group of (twisted or untwisted) Lie type, or one of the above sporadic groups. See [a2] for a discussion of the proof.

References

[a1] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups" , Clarendon Press (1985)
[a2] D. Gorenstein, "Finite simple groups. An introduction to their classification" , Plenum (1982)
How to Cite This Entry:
Sporadic simple group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25776