Define an equivalence relation $\sim$ on $F$ by setting $f'\sim f$ if and only if $f-f'=2g$ for some $g\in F$. Prove that $F/\sim$ is finite if and only if $A$ is finite, and in that case $|F/\sim| = 2^{|A|}$.
Assume $F^{ab}(B)\cong F^{ab}(A)$. If $A$ is finite, prove that $B$ is also, and that $A\cong B$ as sets.
This is a question from Chapter II, section 5 of Aluffi's Algebra: Chapter 0. The second part should follow almost immediately from the first, but I'm having trouble proving the first/even getting started with it.
If I assume that $A$ is finite with cardinality $n$, then $F^{ab}(A)\cong \mathbb{Z}^{\oplus A}=\mathbb{Z}^{\oplus n}$. I also know that every element of $\mathbb{Z}^{\oplus n}$ can be written in the form $\sum_{i=1}^nm_ij(i)$ where $j: A\to \mathbb{Z}^{\oplus n}$ is defined by $j(a_i)=(0,\dots, 0, 1, 0,\dots, 0)\in\mathbb{Z}^{\oplus n}$ where $1$ is in the $i$-th spot. (This of course assumes we order $A=\{a_1, \dots, a_n\}$). If someone could point me in the right direction that would be wonderful.
Taking this question even further in section 7, prove that this relation on $F$ is compatible with the group structure. Determine the quotient $F/\sim$ as a better known group. ***I'm not entirely sure what he means by compatible here?
Thanks in advance