I thought about the following question
What is the index of the free group of rank $k$, denoted $F_k$, in the free group of rank $n$, denoted $F_n$?
Let's say for the moment, $k < n$, and say $[F_n:F_k] = i$. Then by Nielsen-Schreier's theorem, and by the fact that $F_k$ is a subgroup of $F_n$, we have that $F_k$ must be isomorphic to $F_{i(n-1)+1}$. Thus we must have $$k = i(n-1)+1 \iff i = \frac{k-1}{n-1}.$$
So if, for example, we set $n = 7$ and $k = 4$, we must have $i = \frac{3}{6} = \frac{1}{2}$.
To me, this $\frac{1}{2}$ does not make sense at all, I was expecting an integer. What is wrong with the above?
I also have another question. I know that $F_n$ is a subgroup of $F_2$ for each $n \in \mathbb{N}$, but is it true that $F_n$ is a subgroup of $F_k$, for $k > 2$ and $n \in \mathbb{N}$ and in particular when $n > k$?