Finding subgroups of a free group with a specific index 
How many subgroups with index two are there of a free group on two generators? What are their generators?

All I know is that the subgroups should have $(2 \times 2) + 1 - 2 = 3$ generators. 

Let $F$ be free on $a$ and $b$, and let $N$ be a subgroup of index $2$. Then $N$ is normal, and there is a map $\phi:F\to\mathbb Z/2\mathbb Z$ with kernel precisely $N$. What are the possible images of $a$ and $b$? For each possible choice you can describe the corresponding generators of $N$ as a normal subgroup, and with a little extra work, generators as subgroup.
