---
tag: 0C093D1T4M
kind: definition
name: Simple group
msc: ["20E32"]
defines: 0YMYMNQ0QZ
uses: [0W77CVBCHQ]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_group
    locator: lead section
---

## Definition

A [[group|G5S1M9]] $G$ is simple when $G \neq \{e\}$ and the only [[normal subgroups|0CB0DX5BVV]] 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|0JJV80RXHD]].
The [[alternating group|0Z5905Q73D]] $A_n$ 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.