I want to proof that $S_n$ is generated by the set of transpositions ${(1,2),(1,3), \ldots , (1,n)}$ using that $(k,j) = (1,k)(1,j)(1,j)$ but I don't know how to continue. I know this is a easy problem but I dont know what to do.
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$\begingroup$Any permutation $\sigma \in S_n$ can be written as the product of disjoint cycles. Any cycle $(a_1, \dotsc, a_m) \in S_n$ can be written as the product of transpositions by noting $$ (a_1, a_2, a_3,\dotsc, a_m) = (a_m, a_1)(a_{m-1}, a_1)\dotsc(a_3, a_1)(a_2, a_1). $$ Therefore, the set of all transpositions generate $S_n$. Since $$ (1, j)(1, k)(1, j) = (j, k), $$ the set generated by $(1, 2), \dotsc, (1, n)$ contains all transpositions. Therefore, the set generated by $(1, 2), \dotsc, (1, n)$ is $S_n$.
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