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Let $G$ be a group such that $|G|=pq$, where $p$ and $q$ are primes and $p<q$. If $G$ is abelian, then it can be immersed in $S_{p+q}$ but it cannot be immersed in $S_{p+q-1}$

Note: $S_n$ is the group of permutations of n elements.

I am trying to show this result, but couldn't reach anything. I managed to show the following one: Let $G$ be a group such that $|G|=pq$. If $G$ is not abelian, then it can be immersed in $S_{q}$ but it cannot be immersed in $S_{q-1}$

Don't know how to use this to show the previous,tough. Any help would be welcome.

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1 Answer 1

Hint: $G$ must contain an element of order $pq$.

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