Equivalent definitions for a Normalizer of a Subgroup

Question

I've seen some definitions of a Normalizer that do not seem equivalent to me and was wondering if I am missing something.

First to setup the basic stuff:

$H$ is a subgroup of $G$.

Denote the normalizer of $H$ as $N(H)$.

So now the different definitions:

Group 1:

1. $N(H)=\{a \in G: aha^{-1} \in H \text{ for }\forall h \in H\}$
2. $N(H)=\{a \in G: aHa^{-1} \subseteq H\}$

Group 2:

1. $N(H)=\{a \in G: aH=Ha\}$
2. $N(H)=\{a \in G: aHa^{-1}=H\}$
3. $N(H)=\cup\{Ha : aH=Ha\}$
4. $N(H)=\cup\{Ha : aHa^{-1}=H\}$

My conclusions so far:

1. The solutions in Group 1 are equivalent
2. The solutions in Group 2 are equivalent
3. The solutions in Group 1 are not equivalent to those in Group 2 in general.
4. Group 1 basically says to gather all elements for which the conjugate of H is a subgroup of H, while Group 2 says to gather all the elements for which the conjugate of H is equvalent to H.
5. If G is finite, Group 1 will be equivalent to Group 2, because $aHa^{-1}$ is a bijective image of $H$ so if $H$ is finite and $aHa^{-1} \subseteq H$ => $aHa^{-1}=H$.
6. So Group 1 and Group 2 are equivalent if G is finite.

So my question comes to this:

Is this correct or not?

Answer 1

You left one cute one out: the normalizer of $H$ in $G$ is the largest subgroup of $G$ in which $H$ is normal.

Of course, then we have: $H\triangleleft G\iff N(H)=G$.