I have been working with normal subgroups and feel like I am doing something wrong. I understand there are many ways to demonstrate if a subgroup is normal, but the methods seem to take longer than I would expect. Perhaps I simply need to work through more problems and get used to it, but I can't shake the sense in the back of my head.
For a group G and a subgroup H, the methods I have been using to show H is normal are:
$\bullet\;gH=Hg$, $\forall g\in G$
$\bullet\;ghg^{-1}\in H$, $\forall g\in G$ and $\forall h\in H$
$\bullet\;gHg^{-1}=H$, $\forall g\in G$
It seems really tedious to check whether the subgroup is normal, especially if the group (and subgroup) have greater orders. What are some other ways to check whether a subgroup is normal, and are they more efficient?
A subgroup $H \subset G$ is normal in $G$ if and only if $H$ is the kernel of a homomorphism, if and only if the operation $aH*bH:=abH$ is well defined, if and only if for all $a, b \in G$ $aHbH:=\{ah_1bh_2 | h_1, h_2 \in H\}=abH.$ Amongst other things, this last claim follows from the fact that $$ah_1bh_2=ab(b^{-1}h_1b)h_2,$$ and now the expression in brackets is in $H$, so the whole expression is in $abH$. Think of a normal subgroup intuitively as a subgroup you can annihilate(or as my algebra professor would say, 'kill'). By conjugation one can in a sense 'neglect' the $h_1$ that is between $a$ and $b$. If $H$ is the kernel of a homomorphism it literally gets annihilated, as it is sent to the identity element in the image. Then $H$ acts as doing 'nothing' in the image. Maybe this is a bit vage but I have the sense you need some intuition for the notion of normality.