Is a group a normal subgroup of itself?

Question

I know this might sound very stupid, but is a group $G$ a normal subgroup of itself? I would say yes. For a subgroup $H<G$ the following is equivalent that for all $a\in G$:

i) $aH=Ha$

ii) $aHa^{-1}\subset H$

iii) $aHa^{-1}= H$

but for $a\in G$ we have that $aGa^{-1}\subset G$ and $G$ is a subgroup of itself.

Am I right or did I do a mistake somewhere?

Answer 1

Yes, you are right. Every non-trivial group $G$ always has at least two normal subgroups: $G$ itself and $\{e_G\}$.