Namespaces
Variants
Actions

Difference between revisions of "Talk:Sporadic simple group"

From Encyclopedia of Mathematics
Jump to: navigation, search
(table centring)
(Cleaned out Talk page)
Line 1: Line 1:
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–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
 
| $2^{21}.3^3.5.7.11^3.23.29.31.37.43$
 
|-
 
| $Co_1$
 
| 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–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====
 
<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>
 

Revision as of 18:51, 30 April 2012

How to Cite This Entry:
Sporadic simple group. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Sporadic_simple_group&oldid=25780