A simple group is a group with no normal subgroups other than e (the trivial group) and itself. The most common examples of simple groups are the alternating groups A_n, where n > 4.

The classification theorem of finite simple groups states that there are four basic types of finite simple groups, along with twenty-six `sporadic simple groups'. Because of the size (over 15,000 pages) of the proof of this theorem, it is also known as the enormous theorem.