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divisible group

Definition

Let $G$ be a group. For the following this is often assumed to be (though is not necessarily) an abelian group. Hence we write here the group operation with a plus-sign

$$+:G\times G\to G.$$+ : G \times G \to G \,.

For $\mathbb{N}$ the natural numbers, there is a function

$$(-)\cdot (-):\mathbb{N}\times G\to G$$(-)\cdot (-) : \mathbb{N} \times G \to G

which takes a group element $g$ to

$$n\cdot g≔{\underbrace{g+g+\cdots +g}}_{n\mathrm{summands}}.$$n \cdot g \coloneqq \underbrace{g + g + \cdots + g}_{n \; summands} \,.

Definition

A group $G$ is called divisible if for every natural number $n$ (hence for every integer) we have that for every element $g\in G$ there is an element $h\in G$ such that

$$g=n\cdot h.$$g = n \cdot h \,.

In other words, if for every $n$ the ‘multiply by $n$’ map $G\stackrel{n}{\to }G$ is a surjection.

Definition

For $p$ a prime number a group is $p$-divisible if the above formula holds for all $n$ of the form $p^k$ for $k\in \mathbb{N}$.

Remark

There is also an abstract notion of $p$-divisible group in terms of group schemes.

Properties

Equivalent characterization

Proposition

Let $A$ be an abelian group.

Assuming the axiom of choice, the following are equivalent:

1. $A$ is divisible

2. $A$ is injective object in the the category Ab of abelian groups

3. the hom functor ${\mathrm{Hom}}_{\mathrm{Ab}}(-,A):{\mathrm{Ab}}^{\mathrm{op}}\to \mathrm{Ab}$ is exact.

This is for instance in (Tsit-YuenMoRi,Proposition 3.19). It follows for instance from using Baer's criterion.

Stability under various operations

Proposition

The direct sum of divisible groups is itself divisible.

Proposition

Every quotient group of a divisible group is itself divisible.

Examples

Example

The additive group of rational number $\mathbb{Q}$ is divisible. Hence also that underlying the real numbers $ℝ$ and the complex numbers.

Hence:

Example

The underlying abelian group of any $\mathbb{Q}$-vector space is divisible.

Also, by prop. 3,

Example

The quotient groups $\mathbb{Q}/\mathbb{Z}$ and $ℝ/\mathbb{Z}$ are divisible (the latter is also written $U(1)$ (for unitary group) or $S^1$ (for circle group)).

Remark

What is additionally interesting about example 3 is that it provides an injective cogenerator for the category Ab of abelian groups. Similarly, $ℝ/\mathbb{Z}$ is an injective cogenerator.

Counter-Example

The following groups are not divisible:

• the additive group of integers $\mathbb{Z}$.

• the cyclic group ${\mathbb{Z}}_n$ for $n\ge 1\in \mathbb{N}$.

References

• Tsit-Yuen Lam, Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York (1999)