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  "body": "\n## Proof\n\n1. Let $n \\ge 2$, let $S_n$ be the [[symmetric group|0RJMNEN3BJ]], and let $T$ be the set of all [[transpositions|02DTRYKGE8]] in $S_n$.\n2. By definition, the [[subgroup generated by|0RG9SPQSWM]] $T$ is the intersection $\\langle T \\rangle$ of all subgroups of $S_n$ that contain $T$. In particular, $\\langle T \\rangle \\subseteq S_n$.\n3. Let $U$ be a subgroup of $S_n$ with $T \\subseteq U$. A [[subgroup|0N0PXCTYA7]] is closed under products, so by induction on the number of factors, $U$ contains each product of elements of $T$.\n4. By the [[decomposition into transpositions|0XK6FECMW5]], each $\\sigma \\in S_n$ is such a product. So $U = S_n$.\n5. By steps 3 and 4, the only subgroup of $S_n$ that contains $T$ is $S_n$ itself. Hence $\\langle T \\rangle = S_n$.\n\n## Idea\n\nAny subgroup that contains all transpositions contains all their products. Every permutation is such a product, so that subgroup is the whole group.\n",
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