 Primitive group action

A transitive group action is primitive if the only blocks are singletons and the whole set; a block has every translate either equal to it or disjoint from it. Orbits of a normal subgroup form blocks. Hence a nontrivial normal subgroup in a faithful primitive action is transitive. Two-transitivity implies primitivity. This normal-subgroup observation is a hypothesis engine for Iwasawa's simplicity lemma.