A5 is simple

Statement
The alternating group of degree five (denoted $$A_5$$) is a simple group.

Related facts

 * A5 is the simple non-Abelian group of smallest order

Proof using sizes of conjugacy classes
The conjugacy class sizes are $$1, 12, 12, 20, 15$$. A normal subgroup must contain the conjugacy class of size $$1$$, and one or more other conjugacy classes. Thus, the order of any normal subgroup must be a sum of some of these numbers, including the $$1$$. By Lagrange's theorem, the order must also divide the order of the group. But no such sum among these numbers divides $$60$$, other than $$1$$ and $$60$$ themselves.