| We have $|Syl_q(G)|=1+kq$ for some $k$ and $|Syl_q(G)|\mid p$. But, since $p<q$, $k=0$, and so $|Syl_q(G)|=1$, and there is a unique Sylow $q$-subgroup. Since the Sylow $q$-subgroup is unique, it is normal (indeed, \PMlinkname{characteristic}{CharacteristicSubgroup}) in $G$. |
We have $|Syl_q(G)|=1+kq$ for some $k$ and $|Syl_q(G)|\mid p$. But, since $p<q$, $k=0$, and so $|Syl_q(G)|=1$, and there is a unique Sylow $q$-subgroup. Since the Sylow $q$-subgroup is unique, it is normal (indeed, \PMlinkname{characteristic}{CharacteristicSubgroup}) in $G$. |
