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divisible group

Contents

Definition

Let GG 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×GG. + : G \times G \to G \,.

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

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

which takes a group element gg to

ngg+g++g nsummands. n \cdot g \coloneqq \underbrace{g + g + \cdots + g}_{n \; summands} \,.
Definition

A group GG is called divisible if for every natural number nn (hence for every integer) we have that for every element gGg \in G there is an element hGh \in G such that

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

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

Definition

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

Remark

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

Properties

Equivalent characterization

Proposition

Let AA be an abelian group.

Assuming the axiom of choice, the following are equivalent:

  1. AA is divisible

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

  3. the hom functor Hom Ab(,A):Ab opAbHom_{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.

Vector space structure

Proposition

The torsion-free? and divisible abelian groups are precisely the \mathbb{Q}-vector spaces, i.e. if AA is torsion-free and divisible, then it carries a unique vector space structure.

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 \mathbb{R} and the complex numbers.

Hence:

Example

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

Also, by prop. 4,

Example

The quotient groups /\mathbb{Q}/\mathbb{Z} and /\mathbb{R}/\mathbb{Z} are divisible (the latter is also written U(1)U(1) (for unitary group) or S 1S^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, /\mathbb{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 n\mathbb{Z}_n for n1n \geq 1 \in \mathbb{N}.

References

  • Tsit-Yuen Lam, Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York (1999)
Revised on December 25, 2014 13:14:28 by Tobias Fritz (127.0.0.1)