simple group

Simple groups are the indivisible atoms of group theory. A group is simple if you cannot break it apart with any quotient: it has no normal subgroup to collapse except the two trivial ones. Just as every integer factors into primes, every finite group is assembled — via composition series — out of simple groups, so the simple groups are the irreducible building blocks of the entire finite theory.

Formally, a group G is simple if it is nontrivial and its only normal subgroups are {1} and G itself. The abelian simple groups are exactly the cyclic groups Z/pZ of prime order — easy to list. The nonabelian simple groups are vastly richer: the smallest is the alternating group A_5 of order 60, and the alternating groups A_n are simple for all n ≥ 5.

The Classification of Finite Simple Groups, completed across thousands of journal pages, states that every finite simple group is either cyclic of prime order, an alternating group A_n (n ≥ 5), a group of Lie type, or one of 26 sporadic groups (the largest being the Monster, of order about 8 x 10^53). It is one of the monumental achievements of twentieth-century mathematics. A caution: simplicity says a group has no proper quotients, not that it is small or structurally trivial — simple groups can be enormous and intricate.