Let 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 the natural numbers, there is a function
which takes a group element to
A group is called divisible if for every natural number (hence for every integer) we have that for every element there is an element such that
In other words, if for every the ‘multiply by ’ map is a surjection.
For a prime number a group is -divisible if the above formula holds for all of the form for .
There is also an abstract notion of -divisible group in terms of group schemes.
Let be an abelian group.
Assuming the axiom of choice, the following are equivalent:
is divisible
is injective object in the the category Ab of abelian groups
the hom functor 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 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 is divisible. Hence also that underlying the real numbers and the complex numbers.
Hence:
The underlying abelian group of any -vector space is divisible.
The quotient groups and are divisible (the latter is also written (for unitary group) or (for circle group)).
What is additionally interesting about example is that it provides an injective cogenerator for the category Ab of abelian groups. Similarly, is an injective cogenerator.
The following groups are not divisible:
the additive group of integers .
the cyclic group for .
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 June 17, 2022 at 20:51:27. See the history of this page for a list of all contributions to it.