A Quotient of Free Group If $F$ is a free group on a finite set $S$, then the squares in $F$ generate a normal subgroup $N$ and $F/N$ is elementary abelian $2$-group of order $2^{|S|}$.
Let $F$ be free group on infinite set $S$, and $N$ the normal subgroup generated by squares in $F$. 

Q. $F/N$ is infinite direct sum or product of $|S|$ many copies of $\mathbb{Z}_2$?

 A: You could conclude easily by saying that since the words are finite, then you must be in the direct sum and not the direct product as Quang Hoang points out.
But I think the most adequate point of view in general is that of universal properties (it's the most helpful in general cases) : $F$ has the universal property the any function $f:S\to G$ to a group $G$ extends uniquely to a morphism $\tilde{f}: F\to G$. Then the quotient $F/N$ has the property that any group morphism $\tilde{f}: F\to G$ such that $\tilde{f}(a)^2 = 1$ for all $a\in S$ factors uniquely to a morphism $\overline{f}:F/N\to G$.
So $F/N$ has the property that any function $f:S\to G$ such that $f(a)^2= 1$ for all $a\in S$ extends uniquely to a group morphism $\overline{f}: F/N\to G$.
This is exactly the universal property of $\bigoplus_S \mathbb{Z}/2\mathbb{Z}$, since such a function $f:S\to G$ is the same as giving a morphism $f_a: \mathbb{Z}/2\mathbb{Z}\to G$ for all $a\in S$.
So $F/N$ and $\bigoplus_S \mathbb{Z}/2\mathbb{Z}$ have the same universal property : they are canonically equal.
