Sporadic simple group

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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.'

<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


[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 non-Abelian 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.


[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:
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article