I started learning group theory and got stuck with one proof...
A normal subgroup $N \subseteq G$ is defined as: $\forall g \in G; n \in N \space\ (g*n*g^{-1} \in N) $
How can I show that, using this definition, left and right cosets of $N$ in $G$ are equal? Can you please give me some hint?
Let $g\in G$. So you need to show that $gN=Ng$ and you can do that by proving that $gN\subset Ng$ and $Ng\subset gN$. So if $x\in gN$ you need to show that $x\in Ng$. Note that $gng^{-1}\in N$ means that $\exists n'\in N,\,gng^{-1}=n'$.
Remark: you can actually show that the converse is also true, i.e, if all left and right cosets of $N$ in $G$ are equal then $N$ is a normal subgroup of $G$.