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  "body": "\n## Definition\n\nLet a group $G$ [[act|0WCY42EV9V]] on a nonempty set $\\Omega$.\nThe action is *primitive* if:\n\n1. it is [[transitive|02GK6HZYBT]], and\n2. the only [[block systems|SF738NKWJ0]] of $\\Omega$ for $G$ are the partition into singletons and the partition $\\set{\\Omega}$ with one part.\n\nA [[permutation group|ZDJQRKPWGA]] $G \\le \\operatorname{Sym}(\\Omega)$ is *primitive* if its natural action on $\\Omega$ is primitive.\n\n## Remarks\n\nCondition 2 says that every block $B$ of $G$ has $\\abs{B} = 1$ or $B = \\Omega$.\nIf $\\abs{\\Omega} \\ge 2$ and $G$ is transitive, then $G$ is primitive if and only if the [[stabiliser|0AYMP41THN]] $G_\\omega$ of a point $\\omega$ is a maximal subgroup of $G$.\n\nExample: a transitive group $G$ of prime degree $p$.\nFor a transitive group, all parts of a block system have the same size, which divides $p$, so each block system is trivial.\nFor instance, the [[cyclic group|0KVGH0J5B3]] generated by $(1\\,2\\,3\\,4\\,5)$ is primitive on $\\set{1, \\dots, 5}$.\nEvery $2$-transitive group is primitive (see [[multiply transitive group action|BDJ1AX4XBB]]).\n\nNon-example: the eight symmetries of a square acting on its corners $1, 2, 3, 4$ in cyclic order.\nThe action is transitive, but the partition $\\set{\\set{1, 3}, \\set{2, 4}}$ into the two diagonals is a nontrivial block system, so clause 2 fails.\n\nIf $G$ acts primitively and [[faithfully|0P7G9NGG2G]], then every [[normal subgroup|0CB0DX5BVV]] $N \\ne 1$ of $G$ is transitive on $\\Omega$, because the orbits of $N$ form a block system.\n",
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