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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.
+
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;"
 
{| class="wikitable" style="margin: 1em auto 1em auto;"
 
|+ The twenty-six sporadic simple groups
 
|+ The twenty-six sporadic simple groups
! notation
+
! Notation
! name
+
! Name
! order
+
! Order
 
|-
 
|-
 
| $M_{11}$
 
| $M_{11}$
| rowspan="5" | Mathieu groups
+
| rowspan="5" | [[Mathieu group | Mathieu groups]]
| $2^4.3^2.5.11$
+
| $2^4\d 3^2\d 5\d 11$
 
|-
 
|-
 
| $M_{12}$
 
| $M_{12}$
| $2^6.3^3.5.11$
+
| $2^6\d 3^3\d 5\d 11$
 
|-
 
|-
 
| $M_{22}$
 
| $M_{22}$
| $2^7.3^2.5.7.11$
+
| $2^7\d 3^2\d 5\d 7\d 11$
 
|-
 
|-
 
| $M_{23}$
 
| $M_{23}$
| $2^7.3^2.5.7.11.23$
+
| $2^7\d 3^2\d 5\d 7\d 11\d 23$
 
|-
 
|-
 
| $M_{24}$
 
| $M_{24}$
| $2^{10}.3^3.5.7.11.23$
+
| $2^{10}\d 3^3\d 5\d 7\d 11\d 23$
 
|-
 
|-
 
| $J_1$
 
| $J_1$
 
| Janko group
 
| Janko group
| $2^3.3.5.7.11.19$
+
| $2^3\d 3\d 5\d 7\d 11\d 19$
 
|-
 
|-
 
| $J_2$, $HJ$
 
| $J_2$, $HJ$
 
| Hall–Janko group
 
| Hall–Janko group
| $2^7.3^3.5^2.7$
+
| $2^7\d 3^3\d 5^2\d 7$
 
|-
 
|-
 
| $J_3$, $HJM$
 
| $J_3$, $HJM$
 
| Hall–Janko–McKay group
 
| Hall–Janko–McKay group
| $2^7.3^5.5.17.19$
+
| $2^7\d 3^5\d 5\d 17\d 19$
 
|-
 
|-
 
| $J_4$
 
| $J_4$
 
| Janko group
 
| Janko group
| $2^{21}.3^3.5.7.11^3.23.29.31.37.43$
+
| $2^{21}\d 3^3\d 5\d 7\d 11^3\d 23\d 29\d 31\d 37\d 43$
 
|-
 
|-
 
| $Co_1$
 
| $Co_1$
 
| rowspan="3" | Conway groups
 
| rowspan="3" | Conway groups
| $2^{21}.3^9.5^4.7^2.11.13.23$
+
| $2^{21}\d 3^9\d 5^4\d 7^2\d 11\d 13\d 23$
 
|-
 
|-
 
| $Co_2$
 
| $Co_2$
| $2^{18}.3^6.5^3.7.11.23$
+
| $2^{18}\d 3^6\d 5^3\d 7\d 11\d 23$
 
|-
 
|-
 
| $Co_3$
 
| $Co_3$
| $2^{10}.3^7.5^3.7.11.23$
+
| $2^{10}\d 3^7\d 5^3\d 7\d 11\d 23$
 
|-
 
|-
 
| $F_{22}$, $M(22)$
 
| $F_{22}$, $M(22)$
 
| rowspan="3" | Fischer groups
 
| rowspan="3" | Fischer groups
| $2^{17}.3^9.5^2.7.11.13$
+
| $2^{17}\d 3^9\d 5^2\d 7\d 11\d 13$
 
|-
 
|-
 
| $F_{23}$, $M(23)$
 
| $F_{23}$, $M(23)$
| $2^{18}.3^{13}.5^2.7.11.13.17.23$
+
| $2^{18}\d 3^{13}\d 5^2\d 7\d 11\d 13\d 17\d 23$
 
|-
 
|-
 
| $F_{24}^\prime$, $M(24)^\prime$
 
| $F_{24}^\prime$, $M(24)^\prime$
| $2^{21}.3^{16}.5^2.7^3.11.13.17.23.29$
+
| $2^{21}\d 3^{16}\d 5^2\d 7^3\d 11\d 13\d 17\d 23\d 29$
 
|-
 
|-
 
| $HS$
 
| $HS$
 
| Higman–Sims group
 
| Higman–Sims group
| $2^9.3^2.5^3.7.11$
+
| $2^9\d 3^2\d 5^3\d 7\d 11$
 
|-
 
|-
 
| $He$, $HHM$
 
| $He$, $HHM$
 
| Held–Higman–McKay group
 
| Held–Higman–McKay group
| $2^{10}.3^3.5^2.7^3.17$
+
| $2^{10}\d 3^3\d 5^2\d 7^3\d 17$
 
|-
 
|-
 
| $Suz$
 
| $Suz$
 
| Suzuki group
 
| Suzuki group
| $2^{13}.3^7.5^2.7.11.13$
+
| $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$
 
|-
 
|-
 
| $M^c$
 
| $M^c$
 
| McLaughlin group
 
| McLaughlin group
| $2^7.3^6.5^3.7.11$
+
| $2^7\d 3^6\d 5^3\d 7\d 11$
 
|-
 
|-
 
| $Ly$
 
| $Ly$
 
| Lyons group
 
| Lyons group
| $2^8.3^7.5^6.7.11.31.37.67$
+
| $2^8\d 3^7\d 5^6\d 7\d 11\d 31\d 37\d 67$
 
|-
 
|-
 
| $Ru$
 
| $Ru$
 
| Rudvalis group
 
| Rudvalis group
| $2^{14}.3^3.5^3.7.13.29$
+
| $2^{14}\d 3^3\d 5^3\d 7\d 13\d 29$
 
|-
 
|-
 
| $O'N$, $O'NS$
 
| $O'N$, $O'NS$
 
| O'Nan–Sims group
 
| O'Nan–Sims group
| $2^9.3^4.5.7^3.11.19.31$
+
| $2^9\d 3^4\d 5\d 7^3\d 11\d 19\d 31$
 
|-
 
|-
 
| $F_1$, $M$
 
| $F_1$, $M$
 
| Monster, Fischer–Griess group
 
| Monster, Fischer–Griess group
| $2^{46}.3^{20}.5^9.7^6.11^2.13^3.17.19.23.29.31.41.47.59.71$
+
| $2^{46}\d 3^{20}\d 5^9\d 7^6\d 11^2\d 13^3\d 17\d 19\d 23\d 29\d 31\d 41\d 47\d 59\d 71$
 
|-
 
|-
 
| $F_2$, $B$
 
| $F_2$, $B$
 
| Baby monster
 
| Baby monster
| $2^{41}.3^{13}.5^6.7^2.11.13.17.19.23.31.47$
+
| $2^{41}\d 3^{13}\d 5^6\d 7^2\d 11\d 13\d 17\d 19\d 23\d 31\d 47$
 
|-
 
|-
 
| $F_3$, $E$, $Th$
 
| $F_3$, $E$, $Th$
 
| Thompson group
 
| Thompson group
| $2^{15}.3^{10}.5^3.7^2.13.19.31$
+
| $2^{15}\d 3^{10}\d 5^3\d 7^2\d 13\d 19\d 31$
 
|-
 
|-
 
| $F_5$, $D$, $HN$
 
| $F_5$, $D$, $HN$
 
| Harada–Norton group
 
| Harada–Norton group
| $2^{14}.3^6.5^6.7.11.19$
+
| $2^{14}\d 3^6\d 5^6\d 7\d 11\d 19$
 
|}
 
|}
  
====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====
 
====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.
+
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====
 
====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>
+
{|
 +
|-
 +
|  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.
 +
|-
 +
|  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}}
 +
|-
 +
|}

Revision as of 22:42, 30 April 2012

2020 Mathematics Subject Classification: Primary: 20D08 [MSN][ZBL]

A sporadic simple group is 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.

$\def\d{\cdot}$

The twenty-six sporadic simple groups
Notation Name Order
$M_{11}$ 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–Janko group $2^7\d 3^3\d 5^2\d 7$
$J_3$, $HJM$ Hall–Janko–McKay group $2^7\d 3^5\d 5\d 17\d 19$
$J_4$ Janko group $2^{21}\d 3^3\d 5\d 7\d 11^3\d 23\d 29\d 31\d 37\d 43$
$Co_1$ Conway groups $2^{21}\d 3^9\d 5^4\d 7^2\d 11\d 13\d 23$
$Co_2$ $2^{18}\d 3^6\d 5^3\d 7\d 11\d 23$
$Co_3$ $2^{10}\d 3^7\d 5^3\d 7\d 11\d 23$
$F_{22}$, $M(22)$ 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–Sims group $2^9\d 3^2\d 5^3\d 7\d 11$
$He$, $HHM$ Held–Higman–McKay group $2^{10}\d 3^3\d 5^2\d 7^3\d 17$
$Suz$ Suzuki group $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$
$M^c$ 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–Sims group $2^9\d 3^4\d 5\d 7^3\d 11\d 19\d 31$
$F_1$, $M$ Monster, Fischer–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–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. [As], [Go]) has led to the conclusion that 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. A discussion of the proof is given in [Go] up to the uniqueness proof for the monster group $F_1$, which did appear in [GrMeSe].


References

[As] M. Aschbacher, "The finite simple groups and their classification", Yale Univ. Press (1980) MR0555880 Zbl 0435.20007
[CoCuNoPaWi] J.H. Conway, R.T. Curtis, S.P. Norton, R.A. Parker, R.A. Wilson, "Atlas of finite groups", Clarendon Press (1985) MR0827219 Zbl 0568.20001
[Go] D. Gorenstein, "Finite simple groups. An introduction to their classification", University Series in Mathematics. Plenum Publishing Corp., New York (1982) MR0698782 Zbl 0483.20008
[GrMeSe] R.L. Griess, U. Meierfrankenfeld, Y. Segev, "A uniqueness proof for the Monster". Ann. of Math. (2) 130 (1989), no. 3, 567–602.
[Sy] 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 MR0595144 Zbl 0466.20006
How to Cite This Entry:
Sporadic simple group. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25784
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article