I've just solved a basic group theory exercise which is: decide if $\{1,r,r^2,r^3\}$ is a normal subgroup of $\mathbb D_4$ (I mean the dihedral group of $8$ elements, not the one of $4$). I've used the following result: if $H$ is a subgroup of index $2$, then $H \lhd G$. Since $|\mathbb D_4:R|=\dfrac{|\mathbb D_4|}{|R|}=\dfrac{8}{4}=2$, then $ R \lhd \mathbb D_4$.
I was looking for an alternative solution to mine without using the concept of index. For example, a direct proof calculating $(r^js^k)r^p(r^js^k)^{-1}$ for arbitraries $r^p \in R$, $r^js^k \in \mathbb D_4$, or an intuitive idea of why the rotations are invariant under conjugation. Thanks in advance.
The intuition for the subgroup of rotations being normal is actually quite nice in this example, and easy to imagine visually. These are the symmetries of a square, and the element $s$, of order $2$, is a "flip". (Okay, I'm assuming that is what your $s$ denotes.) If you flip a square over, then rotate it, then flip it back again ($s$ followed by some $r^i$ followed by $s^{-1}=s$ again), the net result is the same as if you had just performed a rotation. This shows that the rotation subgroup is invariant under conjugation by $s$. Since it is also, clearly, invariant under conjugation by $r$, and $s$ and $r$ together generate the entire group, it must be normal. This generalises to other dihedral groups too!