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Difference between revisions of "Sporadic simple group"

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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 [[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.
  

Revision as of 18:58, 30 April 2012

2010 Mathematics Subject Classification: Primary: 20D08 [MSN][ZBL]

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://www.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