Assume $H\le S_n$ contains an element of order $n$ and a transposition. Given that $n$ is prime, does $H=S_n$? [duplicate]

Solution 1:

Hint: Say the transposition transposes $1$ and $k+1$, and let $\sigma$ be the cycle $(123\cdots n)$. Then $\sigma^k$ is a cycle (this part uses $n$ prime), so you can relabel the elements so that the transposition is $(12)$. Can you finish from here?

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