Is it true that $S_n$ is generated by the transposition $(12)$ and the set of $3$-cycles $\{(123),(124),\dots ,(12n)\}$ according to the splitting lemma of $Z_2$ and $A_n$?
2026-03-26 16:04:00.1774541040
constructing new genereating set for $S_n$
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Hint: Is $(123\dots n)\in \langle\{(12), (123),\dots, (12n)\}\rangle$, the subgroup of $S_n$ generated by the given set?
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