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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
For $\mathbb{N}$ the natural numbers, there is a function
which takes a group element $g$ to
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
In other words, if for every $n$ the ‘multiply by $n$’ map $G \stackrel{n}{\to} G$ is a surjection.
For $p$ a prime number a group is $p$-divisible if the above formula holds for all $n$ of the form $p^k$ for $k \in \mathbb{N}$.
There is also an abstract notion of $p$-divisible group in terms of group schemes.
Compare also the notion of a divisible module.
Let $A$ be an abelian group.
Assuming the axiom of choice, the following are equivalent:
$A$ is divisible
$A$ is injective object in the the category Ab of abelian groups
the hom functor $Hom_{Ab}(-,A) : Ab^{op} \to Ab$ is exact.
This is for instance in (Tsit-YuenMoRi,Proposition 3.19). It follows for instance from using Baer's criterion.
The torsion-free and divisible abelian groups are precisely the rational vector spaces, i.e. if $A$ is torsion-free and divisible, then it carries a unique vector space structure.
The direct sum of divisible groups is itself divisible.
Every quotient group of a divisible group is itself divisible.
The additive group of rational number $\mathbb{Q}$ is divisible. Hence also that underlying the real numbers $\mathbb{R}$ and the complex numbers.
Hence:
The underlying abelian group of any $\mathbb{Q}$-vector space is divisible.
The quotient groups $\mathbb{Q}/\mathbb{Z}$ and $\mathbb{R}/\mathbb{Z}$ are divisible (the latter is also written $U(1)$ (for unitary group) or $S^1$ (for circle group)).
What is additionally interesting about example is that it provides an injective cogenerator for the category Ab of abelian groups. Similarly, $\mathbb{R}/\mathbb{Z}$ is an injective cogenerator.
The following groups are not divisible:
the additive group of integers $\mathbb{Z}$.
the cyclic group $\mathbb{Z}_n$ for $n \geq 1 \in \mathbb{N}$.
Tsit-Yuen Lam, Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York (1999) (doi:10.1007/978-1-4612-0525-8)
Phillip A. Griffith (1970), Infinite Abelian group theory. Chicago Lectures in Mathematics. University of Chicago Press. ISBN 0-226-30870-7
Last revised on August 20, 2024 at 21:00:15. See the history of this page for a list of all contributions to it.