= Minimal normal subgroup dichotomy for two-transitive groups
A <minimal normal subgroup> of a finite faithful <two-transitive group action> is either regular elementary abelian or primitive nonabelian simple. The imprimitive case reduces to two-point stabilizers being trivial, then counting derangements gives a characteristic regular Sylow subgroup. In the primitive nonregular case, each simple direct factor is transitive. Two commuting transitive factors are regular and are mutual full centralizers. Thus at most two factors occur and $|N|=n^2$. But the common subdegree of $N_\alpha$ then divides both $n$ and $n-1$, forcing $N_\alpha=1$, a contradiction. Only one factor remains.
Back to article page
