I am trying to understand the relation between the action of a group on one another and the order of the groups, here is a statement said to me by my professor:
"if $H_5$ acts on $H_{17}$ by conjugation, this means that $5|16$ which can not happen." where $H_5$ is a group of prime order $5$ and $H_{17}$ is a group of prime order $17.$
I know that $16$ is the order of the group of automorphisms of $H_{17},$ but still, I did not get what my professor said. Could anyone tell me what theorem says what my professor says?
If $H_5$ acts non-trivially on $H_{17}$ by automorphisms, then there is a homomorphism from $H_5$ to $H_{17}^×$, the automorphism group of $H_{17}$, whose order is $16$. Of course, if the kernel is not $H_5$, then it's trivial. There's a couple ways to see that then $5\mid16$. You could use the first isomorphism theorem, but you might as well just note that the order of the image of any element in $H_5$ has to be $5$. But it also has to divide $16$, by Lagrange.