nLab sporadic finite simple group

Idea

The sporadic finite simple groups are the exceptional structures among finite groups:

According to the classification of finite simple groups, there are 18 countably infinite families and 26 sporadic simple groups. The latter groups do not fit into any systematic classification, but there are a number of links between them. For example, the Monster group $M$, the largest of the sporadic groups, contains all but six (the ‘pariah groups’, $J_1, J_3,Ru,ON,Ly,J_4$) of the other sporadic groups as subquotients. These 20 sporadic groups comprise what is termed the ‘Happy Family’ by Robert Griess, (Griess 98).

The Happy Family is taken to be formed of three generations

• First: five subquotients of the Mathieu group $M_{24}$ = symmetries of the binary Golay code
• Second: seven subquotients of the Conway group $Co_1$ = symmetries of the Leech lattice, $mod \{\pm 1\}$.
• Third: eight subquotients of the Monster group = conformal symmetries of the monster vertex operator algebra

Examples

• Conway group

• monster group

• Mathieu group

Related concepts

• Moonshine

• automorphisms of super vertex operator algebras

References

• Robert A. Wilson, The Finite Simple Groups. Springer, Graduate Mathematics series 251 (2009).

• Robert Griess, Twelve Sporadic Groups, Springer, 1998 (doi:10.1007/978-3-662-03516-0)