---
tag: ZXVBEPMTCN
kind: concept
name: Divisible group
msc: [20K40, 20K25]
specializes: [082GHNPJ53]
sources:
  - url: https://en.wikipedia.org/wiki/Divisible_group
same_as:
  wikidata: Q1782332
  wikipedia: Divisible group
---

## Summary

A divisible group is an [[abelian group|082GHNPJ53]] in which every element can be divided by every positive integer.
The rational numbers, the real numbers and $\Q/\Z$ are divisible; the integers are not.
Divisible groups are exactly the [[injective|85H7PRHDPZ]] $\Z$-modules, and every divisible group is a direct sum of copies of $\Q$ and of [[Prüfer groups|F0XAMM2PPS]].
Every abelian group embeds in a divisible group, and a divisible subgroup is always a direct summand.

## Explanations

### L1 human

In $\Q$, every number can be halved, divided by three, and so on, and the answer stays in $\Q$.
A divisible group is an abelian group with this property for every element and every positive integer.
In $\Z$ this fails, since $1$ has no half.
On the circle of angles, every angle can be divided by $n$, so the circle group is divisible too.