Generators of a Dihedral Group

Question

Let $D_4=\{ 1,r,r^2,r^3,s, sr, sr^2, sr^3\}$. I want to show that $<s> $ is a normal subgroup of $<s,r^2>$ but $<s>$ is not a normal subgroup of $D_{4}$.

I think $<s> = \{1,s\}$ and $<s,r^2> = \{1,s, sr^2,r^2 \}$

$<s>$ is a subgroup because it contains the identity element and is a group. I want to show it is a normal subgroup of $<s,r^2>$. So, I should find the left and right cosets.

$$\begin{array}{ll} \mbox{Left Cosets} & \mbox{ Right cosets}\\ 1<s> = \{1,s\} & <s>1 = \{1,s\} \\ s<s> = \{s,1 \} & <s>s = \{s,1\} \\ sr^2<s> = \{sr^2, sr^2s\} & <s>sr^2 = \{sr^2, ssr^2\} = \{sr^2,r^2 \}\\ r^2<s> = \{ r^2,r^2s\} & <s>r^2 = \{r^2,sr^2\} \end{array}$$

By what i've written, the left and right cosets are not exactly the same. What have I done wrong?

Answer 1

Following the discussion in the comments as to where the confusion lies:

$sr^2s=s(r^2s)=s(sr^2)=ssr^2=s^2r^2=r^2$.

Answer 2

Here $s$ and $r$ are not so good in the sense that $sr\neq rs$; but they are not so bad in the sense that we have, $sr=r^3s$ (and $rs=sr^3$). Can you use this one to check whether each left coset is the "next" right coset?