---
tag: 0JJV80RXHD
kind: statement
role: proposition
name: Abelian simple groups have prime order
msc: ["20E32", 20K01]
uses: [0C093D1T4M, 0B85Q41RP6, 0JM7PA288B, 0KB1ND7XDS]
sources:
  - url: https://en.wikipedia.org/wiki/Simple_group
    locator: Examples
  - url: https://en.wikipedia.org/wiki/Simple_group
    locator: Finite simple groups
---

## Statement

Let $G$ be an [[abelian|082GHNPJ53]] [[group|G5S1M9]].
Then $G$ is [[simple|0YMYMNQ0QZ]] if and only if $G$ is finite of prime order.
In that case, $G$ is [[cyclic|0KVGH0J5B3]], and $G \cong \mathbb{Z}/p\mathbb{Z}$ for a prime number $p$.

## Remarks

In an abelian group, each subgroup is normal, so a simple abelian group has no subgroup except $\{e\}$ and itself.
Then $G = \langle g \rangle$ for each $g \neq e$, so $G$ is cyclic; $\mathbb{Z}$ is not simple, because $2\mathbb{Z}$ is a proper nontrivial subgroup.
A finite cyclic group of order $n$ has a subgroup of order $d$ for each divisor $d$ of $n$, so $n$ must be prime.

## Explanations

### L1 human

In an abelian group, all subgroups are normal.
So a simple abelian group has no subgroups at all except the trivial ones.
Only the groups of prime order have this property.