exceptional structures, exceptional isomorphisms
exceptional finite rotation groups:
and Kac-Moody groups:
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
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)
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