$G = S_7$, $a=(2,5,7)$, $b=(5,7)(4,3,1,6)$, $H = \langle a, b \rangle$, and let $K=\langle b \rangle$. Is K normal in H?

Question

$K$ is normal in $H$ if $\forall h \in H, hKh^{-1}\subseteq K$. Alternatively you can check if the right cosets equal the left cosets. I was wondering if there is an easier way of doing this than doing out the computations.

Is there anyother way of checking if the subgroup $K$ is normal than writing out the elements of $H$ and calculating $hKh^{-1}$ to see if it is a subgroup of $K$.

Answer 1

Let's compute: $aba^{-1}=(257)(57)(4316)(275)=(1643)(27)$. But, nothing in $K$ moves $2$. Thus $aba^{-1}\not\in K$ and $K$ isn't normal.