= Even primitive groups of safe-prime degree
{title2=$p=2q+1,\ q\text{ prime}:\ G\le A_p\Longrightarrow G\text{ simple or }|G|=pq$}
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 $G=NN_G(P)$; 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 $|G|=pq$.
Back to article page
