Sporadic simple group

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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.


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 $2^{13}\d 3^7\d 5^2\d 7\d 11\d 13$
$McL$ 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 $2^{14}\d 3^6\d 5^6\d 7\d 11\d 19$


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].


[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. MR1025167 Zbl 0691.20014
[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:
This article was adapted from an original article by V.D. Mazurov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article