Finite simple groups are divided into two categories: infinite families and sporadic groups. Asporadic group is a finite simple group that does not belong to any infinite family generated by ageneral construction. For example, cyclic groups of prime order form one of the 18 infinite families.For every prime number, p, the cyclic group Zp is simple. Among the sporadic groups, the MonsterGroup is the largest, having an order roughly equal to 8.08 × 1053. Of the 26 sporadic groups, theMonster Group contains 20 as subquotients; these sporadic groups are collectively known as theHappy Family. In the late 1970s, mathematicians discovered a surprising relationship between theMonster Group and certain modular functions. In particular, the Fourier expansion of the modularj-function has coefficients corresponding to sums of the dimensions of irreducible representationsof the Monster Group. For example, the first nontrivial coefficient, 196884, can be written as196883 + 1. Here, 196883 is the dimension of the smallest nontrivial irreducible representationof the Monster Group, and 1 is the dimension of the trivial representation. This unexpectedrelationship became known as Monstrous Moonshine and was later proven by Richard Borcherdsin 1992.
