Simple group

If \(N\) is a proper normal subgroup of \(G\), the group splits into two smaller pieces, \(N\) and \(G/N\). A simple group is exactly one where this cannot be done, so it plays the role of the prime numbers: every finite group is built, through a composition series, from simple factors, and the Jordan–Hölder theorem guarantees that these factors are determined up to order.

Among abelian groups, the simple ones are exactly the cyclic groups of prime order \(\mathbb{Z}_p\): in an abelian group every subgroup is normal, and by Lagrange only a prime order rules out intermediate subgroups. Nonabelian examples are far rarer and more remarkable: \(A_n\) is simple for every \(n\ge5\), and \(A_5\), of order \(60\), is the smallest of them. For \(n=3,4\) it fails: \(A_3\) is cyclic of order \(3\) but \(A_4\) contains the Klein four-group as a normal subgroup.

A useful criterion: if \(G\) is simple and \(\varphi\) is a homomorphism out of \(G\), its kernel is normal, so \(\varphi\) is injective or trivial. Dually, \(G/N\) is simple if and only if \(N\) is a maximal normal subgroup, which gives the method for building composition series. A typical mistake is to think that simple means «no proper subgroups»; \(A_5\) has many subgroups, but none is normal.

The classification of finite simple groups, completed after decades of work by hundreds of mathematicians, sorts them into infinite families (cyclic of prime order, alternating, groups of Lie type) and twenty-six sporadic groups. The Sylow theorems, already seen in the course, are the tool used to show that certain orders cannot give simple groups.