Let $H$ be the subgroup of the dihedral group $D_4$ generated by the elements $\left<I\ (\text{identity}),p^2\epsilon\ (\text{flip across the horizontal diagonal})\right>$. I have to tell whether or not this is a normal subgroup of the dihedral group $D_4$ (the regular polygon with 4 sides).
I know that $H = \{I,p^2\epsilon\}$. After that I made the multyplicative tale of D_4 and I saw that that $xH = Hx, \forall x \in D_4$, therefor H is normal in $D_4$. But I'm not sure if this is the way it should be done. I think that there is another way which does not imply the multyplicative table.
a subgroup of order $2$ is normal if and only if its non-identity element is in the center. So you just need to find a permutation that does not commute with the horizontal flip. For example, rotating by $90$ degrees clockwise. So the subgroup is not normal.