What's the difference between the center of a group and a normal subgroup? It seems the definition of the center of a group and a normal subgroup are the same so I'm wondering what the difference is between the two?
A group $H$ is normal in $G$ iff $Hg=gH$ for all $g \in G$.
The center of a group $Z(G) = \{z| \in G$ and for all $g \in G, gz=zg\}$
Those statements seem equivalent to me.
 A: The statements are not equivalent. What you’re missing is that $Hg=gH$ does not imply that $hg=gh$ for all $h\in H$: the set of elements $\{hg:h\in H\}$ can be equal to the set of elements $\{gh:h\in H\}$ without each of the individual products $hg$ and $gh$ being the same. 
For a concrete example of this, let $G=S_3$, the symmetry group of an equilateral triangle; you can see its multiplication table here. Let $H=\{e,d,f\}$; it’s easy to check that $H$ is a subgroup of $G$. Then $aH=\{a,b,c\}=Ha$, but $ad=b\ne c=da$. You can go on to check that $xH=Hx$ for every $x\in G$, so that $H$ is normal in $G$, but none of the elements $a,b$, and $c$ commutes with $d$ or $f$.
A: The center is a normal subgroup, but there are normal subgroups which are different from the center. 
For example consider a cyclic group $\mathbb{Z} /6$, since $\mathbb{Z}/6$ is abelian the definition of the center you gave tells us that $Z(\mathbb{Z}/6) = \mathbb{Z}/6$. However there are also normal subgroups $\mathbb{Z}/2$ and $\mathbb{Z}/3$. 
A: The difference is that $Hg = gH$ means that $ \forall h \in H, \forall g\in G, gh \in Hg$ and $ hg \in gH$. Note that it does not require that $gh = hg$, just that it is in the right coset.
On the other hand, for an element $h \in Z(G),  \forall g \in G, hg = gh$ This is a stronger condition. As such the centre is always a normal subgroup, but not all elements of normal subgroups are in the centre.
A: If $x\in Z(G)$, you have that $g^{-1}xg = x$ for every $g\in G$, whereas if $H$ is normal and $x \in H$, you only have that $g^{-1}xg \in H$.  This is a much weaker condition.
In other words, the center is invariant pointwise under conjugation by $G$, whereas in general normal subgroups are only invariant under conjugation as a whole subgroup.
A: If N is a normal subgroup of G $$ N◁G $$ , then $$ gng‾ ∈N ;∀g∈G $$ or simply $$ gNg‾ = N $$
It follows that for a Normal subgroup it's left and right cosets are same . 
$$ gN = Ng ; \ for \ ∀g∈G $$
It does not however imply  for  some $ n_1 ∈N $ that $$ gn_1 = n_1g $$ .
 It rather implies that there is some $ n_2∈N $ such that 
$$ gn_1 = n_2g $$
Here the Group is  unaffected by conjugation. 
Center of Group $Z(G)$  is defined as set of all elements 
$$ \{s: gs=sg ; ∀g∈G\} $$
And that set is a group in itself thus is a subgroup. 
For center of Group $Z(G)$ and $s∈Z(G)$ it follows that 
$$ gsg– =s $$
Here each element in itself is unaffected by conjugation. 
