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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;" 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-color: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="background-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/s086870/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>
+
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.
  
</td></tr> </table>
+
{| class="wikitable" style="margin: 1em auto 1em auto;"
 +
|+ The twenty-six sporadic simple groups
 +
! notation
 +
! name
 +
! order
 +
|-
 +
| $M_{11}$
 +
| rowspan="5" | Mathieu groups
 +
| $2^4.3^2.5.11$
 +
|-
 +
| $M_{12}$
 +
| $2^6.3^3.5.11$
 +
|-
 +
| $M_{22}$
 +
| $2^7.3^2.5.7.11$
 +
|-
 +
| $M_{23}$
 +
| $2^7.3^2.5.7.11.23$
 +
|-
 +
| $M_{24}$
 +
| $2^{10}.3^3.5.7.11.23$
 +
|-
 +
| $J_1$
 +
| Janko group
 +
| $2^3.3.5.7.11.19$
 +
|-
 +
| $J_2$, $HJ$
 +
| Hall&ndash;Janko group
 +
| $2^7.3^3.5^2.7$
 +
|-
 +
| $J_3$, $HJM$
 +
| Hall&ndash;Janko&ndash;McKay group
 +
| $2^7.3^5.5.17.19$
 +
|-
 +
| $J_4$
 +
| Janko group
 +
| $2^{21}.3^3.5.7.11^3.23.29.31.37.43$
 +
|-
 +
| $Co_1$
 +
| rowspan="3" | Conway groups
 +
| $2^{21}.3^9.5^4.7^2.11.13.23$
 +
|-
 +
| $Co_2$
 +
| $2^{18}.3^6.5^3.7.11.23$
 +
|-
 +
| $Co_3$
 +
| $2^{10}.3^7.5^3.7.11.23$
 +
|-
 +
| $F_{22}$, $M(22)$
 +
| rowspan="3" | Fischer groups
 +
| $2^{17}.3^9.5^2.7.11.13$
 +
|-
 +
| $F_{23}$, $M(23)$
 +
| $2^{18}.3^{13}.5^2.7.11.13.17.23$
 +
|-
 +
| $F_{24}^\prime$, $M(24)^\prime$
 +
| $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$
 +
|-
 +
| $HS$
 +
| Higman&ndash;Sims group
 +
| $2^9.3^2.5^3.7.11$
 +
|-
 +
| $He$, $HHM$
 +
| Held&ndash;Higman&ndash;McKay group
 +
| $2^{10}.3^3.5^2.7^3.17$
 +
|-
 +
| $Suz$
 +
| Suzuki group
 +
| $2^{13}.3^7.5^2.7.11.13$
 +
|-
 +
| $M^c$
 +
| McLaughlin group
 +
| $2^7.3^6.5^3.7.11$
 +
|-
 +
| $Ly$
 +
| Lyons group
 +
| $2^8.3^7.5^6.7.11.31.37.67$
 +
|-
 +
| $Ru$
 +
| Rudvalis group
 +
| $2^{14}.3^3.5^3.7.13.29$
 +
|-
 +
| $O'N$, $O'NS$
 +
| O'Nan&ndash;Sims group
 +
| $2^9.3^4.5.7^3.11.19.31$
 +
|-
 +
| $F_1$, $M$
 +
| Monster, Fischer&ndash;Griess group
 +
| $2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$
 +
|-
 +
| $F_2$, $B$
 +
| Baby monster
 +
| $2^{41}.3^{13}.5^6.7^2.11.13.17.19.23.31.47$
 +
|-
 +
| $F_3$, $E$, $Th$
 +
| Thompson group
 +
| $2^{15}.3^{10}.5^3.7^2.13.19.31$
 +
|-
 +
| $F_5$, $D$, $HN$
 +
| Harada&ndash;Norton group
 +
| $2^{14}.3^6.5^6.7.11.19$
 +
|}
  
 
====References====
 
====References====
<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>
+
<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>
 
 
 
 
  
 
====Comments====
 
====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|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.
+
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 5 letters, a group of (twisted or untwisted) Lie type, or one of the above 26 sporadic groups. See [[#References|[a2]]] for a discussion of the proof.
  
 
====References====
 
====References====
<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>
+
<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>

Revision as of 17:48, 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.

The twenty-six sporadic simple groups
notation name order
$M_{11}$ Mathieu groups $2^4.3^2.5.11$
$M_{12}$ $2^6.3^3.5.11$
$M_{22}$ $2^7.3^2.5.7.11$
$M_{23}$ $2^7.3^2.5.7.11.23$
$M_{24}$ $2^{10}.3^3.5.7.11.23$
$J_1$ Janko group $2^3.3.5.7.11.19$
$J_2$, $HJ$ Hall–Janko group $2^7.3^3.5^2.7$
$J_3$, $HJM$ Hall–Janko–McKay group $2^7.3^5.5.17.19$
$J_4$ Janko group $2^{21}.3^3.5.7.11^3.23.29.31.37.43$
$Co_1$ Conway groups $2^{21}.3^9.5^4.7^2.11.13.23$
$Co_2$ $2^{18}.3^6.5^3.7.11.23$
$Co_3$ $2^{10}.3^7.5^3.7.11.23$
$F_{22}$, $M(22)$ Fischer groups $2^{17}.3^9.5^2.7.11.13$
$F_{23}$, $M(23)$ $2^{18}.3^{13}.5^2.7.11.13.17.23$
$F_{24}^\prime$, $M(24)^\prime$ $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$
$HS$ Higman–Sims group $2^9.3^2.5^3.7.11$
$He$, $HHM$ Held–Higman–McKay group $2^{10}.3^3.5^2.7^3.17$
$Suz$ Suzuki group $2^{13}.3^7.5^2.7.11.13$
$M^c$ McLaughlin group $2^7.3^6.5^3.7.11$
$Ly$ Lyons group $2^8.3^7.5^6.7.11.31.37.67$
$Ru$ Rudvalis group $2^{14}.3^3.5^3.7.13.29$
$O'N$, $O'NS$ O'Nan–Sims group $2^9.3^4.5.7^3.11.19.31$
$F_1$, $M$ Monster, Fischer–Griess group $2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$
$F_2$, $B$ Baby monster $2^{41}.3^{13}.5^6.7^2.11.13.17.19.23.31.47$
$F_3$, $E$, $Th$ Thompson group $2^{15}.3^{10}.5^3.7^2.13.19.31$
$F_5$, $D$, $HN$ Harada–Norton group $2^{14}.3^6.5^6.7.11.19$

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 5 letters, a group of (twisted or untwisted) Lie type, or one of the above 26 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://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=11475
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article