Sporadic simple group

2020 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)