Cohomology and Extensions
The free abelian group on a set is the abelian group whose elements are formal -linear combinations of elements of .
Explicit descriptions of free abelian groups are discussed below.
In terms of formal linear combinations
A formal linear combination of elements of a set is a function
such that only finitely many of the values are non-zero.
Identifying an element with the function which sends to and all other elements to 0, this is written as
In this expression one calls the coefficient of in the formal linear combination.
For Set, the group of formal linear combinations is the group whose underlying set is that of formal linear combinations, def. 2, and whose group operation is the pointwise addition in :
The free abelian group on is, up to isomorphism, the group of formal linear combinations, def. 3, on .
For a set, the free abelian group is the direct sum in Ab of -copies of with itself:
Assuming the axiom of choice, then every subgroup of a free abelian group (def. 1) is itself a free abelian group.
(e.g. Lang 02, Appendix 2 §2, page 880) For a full proof see at principal ideal domain this theorem.
- Serge Lang Algebra, Graduate Texts in Mathematics 211 (Revised third ed.), Springer 2002