if H is the only subgroup of G that has order k(G may have many subgroups..),prove that H is normal in G....
Thank you!
By definition: $\displaystyle H$ is normal in $\displaystyle G$ if and only if, for all $\displaystyle g\in G$ we have $\displaystyle gHg^{-1}=H$
Now, show that, for any $\displaystyle g\in G$ , $\displaystyle gHg^{-1}$ is in fact a subgroup of $\displaystyle G$ that has order $\displaystyle
\left| {gHg^{ - 1} } \right| = \left| H \right|
$. Then the rest will follow since $\displaystyle H$ is the only subgroup of that order in $\displaystyle G$.