I have the following class equation:
$$20 = 1 + 4 + 5 + 5 + 5$$
I know that there are subgroups of order $4$ and $5$ in $G$. I see this because $\vert G \vert = \vert \text{centralizer} \vert \cdot \vert \text{conjugacy class} \vert$ and the centralizer is a subgroup of $G$. So, we have $\vert \text{centralizer} \vert = 4$ and $\vert \text{centralizer} \vert = 5$, so there are subgroups of order $4$ and $5$ in $G$. However, I have been told that the subgroup of order $4$ is not normal and the subgroup of order $5$ is normal in $G$. I know that a normal subgroup is the union of conjugacy classes. Is there a way I can easily show their normality? Thank you.
Best Answer
Use Sylow-Theorem to show normality.$|G|=20=2^2*5$.Hence to get $n_5,1+5k/4,for k=0,1,2...$this means $n_5=1$.Hence G has normal subgroup of order 5.