(no subject)

Wow, I didn't do anything today -nyo.

How's it going?

Umm~ nyu my proof before that "every subgroup of a free (abelian) group is free" was wrong, sorry ^^; but! there's one on wikipedia which works? >_>

Here's a proof for free groups with a finite index set.

Let ; let H be a subgroup.

Let for 1 ≤ j ≤ n; it's a subgroup of H.

The subgroup {x ∈ Z | there's an element of H_j with j-th coordinate x} is equal to q_jZ for some q_j. If q_j ≠ 0 then set a_j to be some element with j-th coordinate q_j, otherwise set a_j = 0.

Okay, then I claim that {a_j | a_j ≠ 0} is a basis for H.

Proof. Exercise. :)