Simple Group

A simple group is a group whose only normal subgroups are the trivial subgroup and the whole group.

Examples:

• $\mathbb{Z}_p$ for prime $p$
• $A_5$ as the classical first nonabelian example

Nonexample:

• $A_4$, which has a normal Klein four subgroup

One-line intuition

A simple group cannot be broken into a smaller normal quotient structure.