Give at most a two-sentence synopsis of the proof of Theorem 15.18.

Data from Theorem 15.18.

M is a maximal normal subgroup of G if and only if G/M is simple. 

Proof: Let M be a maximal normal subgroup of G. Consider the canonical homomorphism γ : G → G/M given. Now γ^-1 of any nontrivial proper normal subgroup of G/M is a proper normal subgroup of G properly containing M. But M is maximal, so this can not happen. Thus G/M is simple.

if N is a normal subgroup of G properly containing M, then γ[ N] is normal in G/M. If also N ≠ G, then 

γ[N] ≠ G/M and γ[N] ≠ {M}. 

Thus, if G/M is simple so that no such γ [N] can exist, no such N can exist, and M is maximal.