Simple group

Definition

A group $G$ is simple when $G \neq \{e\}$ and the only normal subgroups of $G$ are $\{e\}$ and $G$.

Remarks

The cyclic groups $\mathbb{Z}/p\mathbb{Z}$ of prime order $p$ are simple, and they are the only abelian simple groups. The alternating group $A_n$ is simple for $n \geq 5$, and $A_5$ is the smallest simple group that is not abelian. A simple group has only the trivial quotient and itself as quotients, so it cannot be split into smaller pieces by a quotient.

Explanations

L1 human

A simple group cannot be broken into a normal subgroup and a quotient group. In this way, simple groups are like the prime numbers among the integers. Each finite group is built from simple groups, by the Jordan–Hölder theorem.