= Prime-degree transposition criterion
{title2=$G=S_p$}
A transitive subgroup $G\le S_p$ for prime $p$ that contains a transposition is $S_p$. Form a <graph> on the $p$ letters with edges given by all conjugates in $G$ of that transposition. The transitive action permutes connected components, so their common size divides $p$. An edge rules out size one, hence the graph is connected. Its edge transpositions generate $S_p$: successive swaps along a path generate the swap of its endpoint with any chosen start. All these edge transpositions belong to $G$.
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