Is a subgroup of a free abelian group free abelian?

It's well-known that that a subgroup of a free group is free. Is a subgroup of a free abelian group (may not be finitely generated) always a free abelian group?

• I've downvoted this, not because it's a bad question, but because it's answered by the first Google hit for "free abelian group."
– JSE
Oct 30 '09 at 5:41
• @JSE: So it is. Oct 30 '09 at 5:43
• More generally every submodule of a free module over a PID is free.
– YCor
Jan 27 '19 at 17:13

A variety of groups $V$ is said to have the Schreier property if every subgroup of a free group in the variety is free. It is a classical theorem of Peter Neumann and James Wiegold that the only varieties of groups with the Schreier property are: the (absolutely) free groups, the free abelian groups, and the free exponent $p$ abelian groups for $p$ prime.
• Grammar nit-pick: you mean, "the only varieties of groups with the Schreier properties are the variety of all groups, the variety of all abelian groups, and the variety of all abelian groups of exponent $p$, for $p$ a prime." Oct 17 '10 at 6:09