Simple Group

A simple group is a group $G$ with $G\ne \{e\}$ such that the only normal subgroups of $G$ are the trivial subgroup $\{e\}$ and $G$ itself.

Simple groups play the role of “building blocks” for general groups via composition series and uniqueness results such as the Jordan–Hölder theorem .

Examples:

• Any cyclic group of prime order is simple.
• The alternating group $A_5$ is a finite nonabelian simple group.
• (Non-example) $\mathbb{Z}$ is not simple: for instance, $2\mathbb{Z}$ is a nontrivial normal subgroup.