= A cycle and a transposition generate the symmetric group exactly at coprime separation
For $n\geq2$ and $1\leq k<n$, an $n$-cycle $c=(1\,2\,\ldots\,n)$ and $t=(1\,1+k)$ generate $S_n$ exactly when $\gcd(n,k)=1$. Their conjugate <transpositions> connect labels differing by $k$ modulo $n$. This <graph> is connected exactly at <coprime> separation, so <transpositions on a connected graph generate the symmetric group>. Otherwise residue classes modulo $\gcd(n,k)$ form a nontrivial <block system> preserved by both generators.
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