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Nielsen–Schreier theorem

{c}

If $H$ has finite index $d$ in the free group $F_n$, then
$$
\boxed{\operatorname{rank}(H)=1+d(n-1).}
$$
Indeed, the $d$-sheeted <covering graph> of the rose with $n$ petals has $d$ vertices and $dn$ edges, so its <fundamental group> has rank $dn-d+1$.

Nielsen–Schreier formula

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 1 / a / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 1 / b / Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 143 / 2 / d / Solution

Nielsen–Schreier formula

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Nielsen–Schreier formula

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