Confusion with Centers, Conjugacy Classes, and Normal Subgroups Ok this is a bit of a soft question, but I am in my first semester of algebra and getting continually confused by (what seems to me) some competing sets.
Let $G$ be a group

The center of $G$ is $Z(G)=\{z\in G: zgz^{-1}=g\ \ \  \forall g\in G\}$



The conjugacy class of $n\in G$ is $[n]=\{a\in G:\exists g\in G\  \text{where}\ gag^{-1}=n\}$



A normal subgroup, $N\subseteq G$, has the property $gng^{-1}\in N$ for all $g\in G$


I understand the definitions, and I see that theorems often reference these three sets. I just don't get where they come from and why they all have this same form of $aba^{-1}$. The only name that makes some sense to me is conjugacy class. Where do "normal" and "center" come from?
My professor just defined these sets one day and they routinely pop up and every time I have to look back at the definition. I try to understand where definitions and names come from, but I just haven't been able to do that here.


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*My question is: where do the names for these sets come from,what is their significance, and how are they related?

 A: If $xy=yx$ for two elements $x,y\in G $ they are said to commute. In a group where any two elements commute (commutative group) is called an abelian group in honour of N.H.Abel. (Let me say these two elements are friendly)
The three definitions you have given are attempts to quantify how much a group is deviating from this abelianness. 
It is still possible that a few elements are friendly with all elements, that is they commute with all. Such elements make up the centre $Z(G)$ of the group. This is a subgroup.  If the group is abelian $Z(G)$ would be the whole of $G$, the smaller this $Z(G)$ farther the group is removed from abelianness.
An alternative way of saying  $xy=yx$ is  that $yxy^{-1}=x$. When abelianness fails $yxy^{-1}$ may not be $x$, but still it can be  considered somewhat close to $x$, called a conjugate of $x$. Fix an $x$, and vary the $y$, you get all conjugates of $x$; this set is called the cojugacy class of $x$. More number of conjugates an element $x$ has more it contributes to non-abelianness.
A normal subgroup is a subgroup which contains all the conjugates of any of its elements.
In an abelian group $Z(G)$ will be whole of $G$; no two elements can be conjuagte; all subgroups will be normal subgroups.
Hope this clarifies.
