# Thread: Another problem in homomorphism and factor groups.....

1. ## Another problem in homomorphism and factor groups.....

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!

2. By definition: $H$ is normal in $G$ if and only if, for all $g\in G$ we have $gHg^{-1}=H$

Now, show that, for any $g\in G$ , $gHg^{-1}$ is in fact a subgroup of $G$ that has order $
\left| {gHg^{ - 1} } \right| = \left| H \right|
$
. Then the rest will follow since $H$ is the only subgroup of that order in $G$.