A simple group is a group whose only normal subgroups are the trivial subgroup of order one and the improper subgroup consisting
of the entire original group. Simple groups include the
infinite families of alternating groups of degree
, cyclic
groups of prime order, Lie-type groups, and
the 26 sporadic groups.
Since all subgroups of an Abelian group are normal and all cyclic groups are Abelian, the only simple cyclic groups are those which have no subgroups other than the trivial subgroup and the improper subgroup consisting of the entire original group. And since cyclic groups of composite order can be written as a group direct product of factor groups, this means that only prime cyclic groups lack nontrivial subgroups. Therefore, the only simple cyclic groups are the prime cyclic groups. Furthermore, these are the only Abelian simple groups.
In fact, the classification theorem of finite groups states that such groups can be classified completely into these five types:
1. Cyclic groups of prime group order,
2. Alternating groups of degree at least five,
4. Lie-type twisted Chevalley groups or the Tits group, and
5. The sporadic groups.
Burnside's conjecture states that every non-Abelian finite simple group has even group order.
The finite (cyclic) group 
forms the subject for the humorous a capella song "Finite
Simple Group (of Order 2)" by the Northwestern University mathematics department
a capella group "The Klein Four."
Renteln and Dundes (2005) give a (rather bad) mathematical joke involving simple groups:
Q: What is purple and all of its offspring have been committed to institutions? A: A simple grape: it has no normal subgrapes.
