A
simple finite group that does not belong to any of the known infinite series of simple finite groups. The twentysix sporadic simple groups are listed in the following table.'
<tbody> </tbody> notation  name  order         Mathieu groups          Janko group   ,  Hall–Janko group   ,  Higman–Janko–McKay group    Janko group   ,    ,  Conway groups   ,    ,    ,  Fischer groups   ,     Higman–Sims group   ,  Held–Higman–McKay group    Suzuki group    McLaughlin group    Lyons group    Rudvalis group   ,  O'Nan–Sims group   ,  Monster, Fischer–Griess group   ,  Baby monster   , ,  Thompson group   , ,  Harada–Norton group  

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) 
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 nonAbelian finite simple group is isomorphic to: an alternating group on at least letters, a group of (twisted or untwisted) Lie type, or one of the above 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=25775