Classification of Finite Simple Groups

Theorem
The finite simple groups can be classified as follows:

Abelian Simple Groups
The Abelian simple groups are the cyclic groups of prime order.

Alternating Groups
The alternating groups $A \left({n}\right)$ for $n \ge 5$.

Groups of Lie Type
The various families of groups of Lie type.

Sporadic Groups
The various sporadic groups: