# [Math] Generators of permutation group

abstract-algebrapermutations

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.

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 $S_n$.