this is a question from Aluffi's Algebra: Chapter 0, Exercise II.5.10. The full question is the following:
Given a free abelian group over a set $A$, denoted $F=F^{ab}(A)$, we define an equivalence relation $\sim$ by $f\sim f'$ 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.
For the first part - assuming $A$ is finite, I have no trouble showing $F/\sim$ is finite. I used the fact that since $F$ is finite $F\cong\mathbb{Z}^{\oplus n}$ where $|A|=n$. So looking at the elements in $F$ as elements in $\mathbb{Z}^{\oplus n}$ we can think of the equivalence relation as $(x_1,\ldots,x_n)\sim (y_1,\ldots,y_n)$ if and only if $x_i\cong y_i\pmod{2}$ for each $i$. From this we can go on to show $F/\sim\cong(\mathbb{Z}/2\mathbb{Z})^{\oplus n}$. I know there are still details to fill in but this is the basic idea of what I have so far.
Showing the other direction is where I am having trouble. It seems that the way to go would be to assume $A$ is infinite and show $F/\sim$ is infinite. Another theorem from the book more generally says that for any set $A$ we have $F\cong\mathbb{Z}^{\oplus A}$. My gut tells me I should be using this fact but I am lost as to how to proceed. I would appreciate any hints to point me in the right direction - thank you very much in advance.
Edit: Using the hint given by Rushy -
Suppose $A$ is infinite. From the text we are given that $F\cong\mathbb{Z}^{\oplus A}$ and that each element in $\mathbb{Z}^{\oplus A}$ can be written in the form $\sum_{a\in A} m_aj_a$ where $m_a\neq 0$ for finitely many $a$. Note that $j_a:A\to\mathbb{Z}$ is defined by $j_a(x)=1$ if $x=a$ and $0$ otherwise. Now, let $a$ and $b$ be distinct elements of $A$ and suppose towards contradiction that $a\sim b$. Then $a-b=2g$ for some $g\in F$. Using our isomorphism above this is equivalent to saying $j_a-j_b=2(\sum_{i=1}^nm_{a_i}j_{a_i})$ where each $m_{a_i}\neq 0$ and each $a_i$ is distinct. Then $(j_a-j_b)(a)=[2(\sum_{i=1}^nm_{a_i}j_{a_i})](a)$ implies that either $1=2m_{a_k}$ where $1\leq k\leq n$ or $1=0$, which is a contradiction in either case. So, $a$ and $b$ are in distinct equivalence classes and since there are infinitely many elements in $A$ it follows that $F/\sim$ must be infinite.