fix: statement and hint of problem 16E

[jfab20]

First of all added the condition for $n \ge 2$ so that there is a prime that divides $|G|$.

More importantly, I think there is a mistake on the hint. You say that you should consider $G$ acting on the left cosets of $H$. This is one of possible solutions, but I don't think you meant that one, because the solutions in which you consider $G$ as the group that acts on cosets, you conclude by getting a morfism from $G$ to $S_p$ and do something with factorias. You also consider the orbit of $H$ for some reason and apply the orbit stabilizer theorem to deduce something about its orbit, but you obviously don't need this because the orbit of $H$ under this particular action gives you all of the cosets (to get $g H$ just multiply by $g$). Finally, in the solution you provided, the group that acts on cosets is $H$, so you probably got confused while writing the solution.

So basically, I added $n \ge 2$ for the order of $G$, changed the hint to "Let $H$ act on left cosets" and then more or less indicated the path of the solution from there. The point is that you use the orbit stabilizer theorem on all orbits to show that either there is only one orbit or all orbits have size 1. Since the first case cannot happen (because the orbit of $H$ is ${H}$ which is not all the cosets) then all orbits must have size 1 and thus $h g H = g H$ for all $h \in H$ and $g \in G$, which implies that $H$ is normal.

[github-actions]

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