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Let $G$ be a transitive subgroup of $S_n$ generated by a transposition and a cycle of order $p$ whith $p$ a prime and $\frac{n}{2} < p < n$. Prove that $G=S_n$. Please ,I would like for the moment just a hint to tackle the problem.

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We have to show that the group operation for $G$ is an equivalence relation, and the main thing is to show is that the number of equivalence classes is $1$. Since we have

Order of $G$ =(size of an equivalence class, say $M$)$\times$(number of equivalence class, say $N$).

i.e. $n=MN$. Now, we must show that size of any equivalence class is same and $N=1$. For the latter half, one can see whether an orbit of $G$ with size $p$ exists or not.

Or, alternatively,

take a look on page 2 and page 4.

http://www.math.uconn.edu/~kconrad/blurbs/galoistheory/galoisSnAn.pdf

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