View source for Sporadic simple group

{{MSC|20D08}}
{{TEX|done}}

A ''sporadic simple group'' is
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.

$\def\d{\cdot}$
{| class="wikitable" style="margin: 1em auto 1em auto;"
|+ The twenty-six sporadic simple groups
! Notation
! Name
! Order
|-
| $M_{11}$
| rowspan="5" | [[Mathieu group | Mathieu groups]]
| $2^4\d 3^2\d 5\d 11$
|-
| $M_{12}$
| $2^6\d 3^3\d 5\d 11$
|-
| $M_{22}$
| $2^7\d 3^2\d 5\d 7\d 11$
|-
| $M_{23}$
| $2^7\d 3^2\d 5\d 7\d 11\d 23$
|-
| $M_{24}$
| $2^{10}\d 3^3\d 5\d 7\d 11\d 23$
|-
| $J_1$
| Janko group
| $2^3\d 3\d 5\d 7\d 11\d 19$
|-
| $J_2$, $HJ$
| Hall&ndash;Janko group
| $2^7\d 3^3\d 5^2\d 7$
|-
| $J_3$, $HJM$
| Hall&ndash;Janko&ndash;McKay group
| $2^7\d 3^5\d 5\d 17\d 19$
|-
| $J_4$
| Janko group
| $2^{21}\d 3^3\d 5\d 7\d 11^3\d 23\d 29\d 31\d 37\d 43$
|-
| $Co_1$
| rowspan="3" | Conway groups
| $2^{21}\d 3^9\d 5^4\d 7^2\d 11\d 13\d 23$
|-
| $Co_2$
| $2^{18}\d 3^6\d 5^3\d 7\d 11\d 23$
|-
| $Co_3$
| $2^{10}\d 3^7\d 5^3\d 7\d 11\d 23$
|-
| $F_{22}$, $M(22)$
| rowspan="3" | Fischer groups
| $2^{17}\d 3^9\d 5^2\d 7\d 11\d 13$
|-
| $F_{23}$, $M(23)$
| $2^{18}\d 3^{13}\d 5^2\d 7\d 11\d 13\d 17\d 23$
|-
| $F_{24}^\prime$, $M(24)^\prime$
| $2^{21}\d 3^{16}\d 5^2\d 7^3\d 11\d 13\d 17\d 23\d 29$
|-
| $HS$
| Higman&ndash;Sims group
| $2^9\d 3^2\d 5^3\d 7\d 11$
|-
| $He$, $HHM$
| Held&ndash;Higman&ndash;McKay group
| $2^{10}\d 3^3\d 5^2\d 7^3\d 17$
|-
| $Suz$
| Suzuki group
| $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$
|-
| $McL$
| McLaughlin group
| $2^7\d 3^6\d 5^3\d 7\d 11$
|-
| $Ly$
| Lyons group
| $2^8\d 3^7\d 5^6\d 7\d 11\d 31\d 37\d 67$
|-
| $Ru$
| Rudvalis group
| $2^{14}\d 3^3\d 5^3\d 7\d 13\d 29$
|-
| $O'N$, $O'NS$
| O'Nan&ndash;Sims group
| $2^9\d 3^4\d 5\d 7^3\d 11\d 19\d 31$
|-
| $F_1$, $M$
| Monster, Fischer&ndash;Griess group
| $2^{46}\d 3^{20}\d 5^9\d 7^6\d 11^2\d 13^3\d 17\d 19\d 23\d 29\d 31\d 41\d 47\d 59\d 71$
|-
| $F_2$, $B$
| Baby monster
| $2^{41}\d 3^{13}\d 5^6\d 7^2\d 11\d 13\d 17\d 19\d 23\d 31\d 47$
|-
| $F_3$, $E$, $Th$
| Thompson group
| $2^{15}\d 3^{10}\d 5^3\d 7^2\d 13\d 19\d 31$
|-
| $F_5$, $D$, $HN$
| Harada&ndash;Norton group
| $2^{14}\d 3^6\d 5^6\d 7\d 11\d 19$
|}


====Comments====
The classification of the finite simple groups (cf. {{Cite|As}}, {{Cite|Go}}) has led to  the  conclusion that 
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. 
A discussion of the proof is given in {{Cite|Go}} up to the uniqueness
proof for the monster group $F_1$, which did appear in {{Cite|GrMeSe}}.


====References====
{|
|-
|  valign="top"|{{Ref|As}}||valign="top"|  M. Aschbacher,     "The finite simple groups and their classification", Yale Univ. Press    (1980)  {{MR|0555880}}  {{ZBL|0435.20007}} 
|-
|  valign="top"|{{Ref|CoCuNoPaWi}}||valign="top"|  J.H. Conway,     R.T. Curtis,   S.P. Norton,   R.A. Parker,   R.A. Wilson,   "Atlas of   finite groups", Clarendon Press    (1985)  {{MR|0827219}}  {{ZBL|0568.20001}} 
|-
|  valign="top"|{{Ref|Go}}||valign="top"|  D. Gorenstein,     "Finite simple groups. An introduction to their classification",  University Series in Mathematics. Plenum Publishing Corp., New York (1982)  {{MR|0698782}}  {{ZBL|0483.20008}} 
|-
| valign="top"|{{Ref|GrMeSe}}||valign="top"| R.L. Griess, U. Meierfrankenfeld, Y. Segev, "A uniqueness proof for the Monster". ''Ann. of Math.'' (2) '''130''' (1989), no. 3, 567–602. {{MR|1025167}} {{ZBL|0691.20014}}
|-
|  valign="top"|{{Ref|Sy}}||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   {{MR|0595144}} {{ZBL|0466.20006}} 
|-
|}