This might be obvious but it is the only thing in the proof that I can't realize. The proof is here: http://groupprops.subwiki.org/wiki/Proper_and_normal_in_quasisimple_implies_central and my question is why the image of $N$ precisely is $NZ(G)/Z(G)$ as it says in 1.
image of a normal subgroup
1
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group-theory