# Talk:Sporadic simple group

From Encyclopedia of Mathematics

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.

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=25779