I apologize in advance for the size of the images I've devoted a lot of time and effort to draw them on the computer and i didn’t manage to re-size. I'd be really thankful if anyone would edit this post and re-size them.
I have this image in my head of a flower whenever I'm thinking about the class equation and I’d like to make sure that there's nothing wrong with it.
For this first diagram let $G$ be a group and let $x, y, z \in G$. I'll denote the normalizers of $x, y$ and $z$ as $N(x), N(y), N(z)$ respectively and the center as the usual $Z(G)$.
There's a normalizer for every element of the group (I've put only 3 for convenience). diagram shows that the center is contained in every normalizer. Additionally, If i would to draw the normalizers of all the elements in $G$ their intersection would be precisely $Z(G)$ (is that right?).
For the second diagram imagine we pick one of the elements of $G$, say x, and look at the partition of $G$ to cosets of $N(x)$.
I've chosen for convenience that the index of $N(x)$ will be $5$. The unique cosets of $N(x)$ are represented as $aN(x), bN(x), cN(x)$ and $dN(x)$. The arrows represent conjugation relation. The unique conjugates of $x$ are given by conjugation with the elements $a, b, c, d$.
ADDED: As PavelC mentioned in the comments there is a mistake in the labelling of the cosets.
No two unique conjugates can be in the same coset and The fact that no element of the center is a conjugate of any other element (apart from itself) can be seen as a consequence of the center being contained in all the normalizers (or maybe more obvious than that is the fact that the normalizer of each of them is the whole group)
Is there a mistake in anything I’ve written?
Apparently everything is correct except the naming of the cosets.
If $cxc^{-1} \in cN(x)$ then $xc^{-1} \in N(x)$ and $c^{-1} \in N(x)$. This is a contradiction since we have to have $c \notin N(x)$ to get a non-trivial coset.
The cosets were supposed to be labelled differently. A correct labelling that agrees with the diagrams can be given by $cxc^{-1}N(x)$ (instead of $cN(x)$).