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In my lecture notes of algebraic number theory they use the following claim to prove that the Galois group of $X^4 +X+1$ is the whole $S_4$

If $K$ is a subgroup of $S_4$ which contains a 3-cycle and a 4-cycle then $K = S_4$.

Could you tell me if this is true and why does it work?

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up vote 2 down vote accepted

Since $K$ contains an element of order 3 and 4, 12 must divide the order of $K$. Since $S_4$ has order 24, if $K$ is not all of $S_4$, then $K$ must have order 12. If we can show that $A_4$ is the only subgroup of order 12 of $S_4$, then we must have $K = A_4$, which is false since it contains an odd cycle.

An argument for this can be found here.

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