nLab Happy Family

Idea

What has been called the Happy Family in Griess 82 is the finite set of those 20 of the 26 sporadic finite simple groups, which are subquotients of the monster group, in contrast to the remaining 6 pariah groups.

The Happy Family is taken to be formed of three generations:

• First: five subgroups of the Mathieu group $M_{24}$ = symmetries of the binary Golay code, namely the Mathieu groups.
• Second: seven subquotients of the Conway group $Co_0$ = symmetries of the Leech lattice, including the three simple Conway groups, $Co_1, Co_2, Co_3$, and the Hall-Janko, Suzuki, McLaughlin and Higman-Sims groups.
• Third: eight subquotients of the Monster group = conformal symmetries of the monster vertex operator algebra, including the Fischer groups and the Thompson sporadic group.

It has been suggested (Lentner 12) that these three generations are generated by an iterated process of forming the centralizer of an involution of a simple group, starting out from the simple group of Lie-type $SL_3(4)$.

References

• Robert Griess, Jr. The Friendly Giant, Invent. Math. 69 (1982) 1-102 (doi:10.1007/BF01389186)

• Simon Lentner, Response to “Why are the sporadic simple groups HUGE?”, MO answer.

See also

• Wikipedia, Monster group