View source for Talk:Sporadic simple group

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.

{| 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====
<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====
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====
<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>