I'm trying to understand normal subgroups and kernels of homomorphisms. Normal subgroups are defined as such:
$gHg^{-1}=H~~~\forall g \in G$
While i was trying to see which subgroups are normal, to verify the above statment, i should run through all elements of $G$.
To find a shorter way, I come up with the below:
$hGh^{-1}\cap H=\emptyset~~~\forall h \in H $ (wrong)
$h(G \setminus H)h^{-1}\cap H=\emptyset~~~\forall h \in H $ (correct)
This is basically telling that, if H is normal, then $hGh^{-1}$ is in $G \setminus H$.
Assume the contrary,
$h_{0}Gh_{0}^{-1}=h_{1}$ implies $h_{0}G=h_{3}$ implies $G=h_{4}$ or $G=H$ which is a contradiction.
Could anybody prove the above statement from the first, or possible vice versa?
Second one cheaper since H has fewer elements to run through, so i would prefer to write tests over the second one.
Regards.