Cohomology and Extensions
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 .
This is for instance in (Tsit-YuenMoRi,Proposition 3.19). It follows for instance from using Baer's criterion.
Vector space structure
The torsion-free? and divisible abelian groups are precisely the -vector spaces, i.e. if is torsion-free and divisible, then it carries a unique vector space structure.
Stability under various operations
The direct sum of divisible groups is itself divisible.
Every quotient group of a divisible group is itself divisible.
The underlying abelian group of any -vector space is divisible.
Also, by prop. 4,
The following groups are not divisible:
- 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