Even primitive groups of safe-prime degree ($p=2q+1,q\text{ prime}:G\le A_p⟹G\text{ simple or }|G|=pq$)

Every nontrivial normal subgroup of a faithful primitive group of prime degree is transitive and contains a Sylow $p$-subgroup $P$. Its normalizer in $A_p$ has order $pq$. For a proper normal subgroup $N$, the Frattini argument gives $G=N{N}_G(P)$; unless $G=C_p$, its $P$-normalizer is the full order-$pq$ normalizer, while $N_N(P)=P$. The Burnside transfer theorem gives a characteristic normal $p$-complement in $N$. This complement is normal in $G$ and cannot be transitive because its order is prime to $p$, so it is trivial. Thus $N=P$ and $|G|=pq$.