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  "body": "\n## Definition\n\nLet a group $G$ [[act|0WCY42EV9V]] on a set $\\Omega$.\nA *block* of $G$ is a nonempty subset $B \\subseteq \\Omega$ such that, for every $g \\in G$, either $gB = B$ or $gB \\cap B = \\emptyset$.\nA *block system* of $G$ is a partition $\\mathcal{P}$ of $\\Omega$ such that $gB \\in \\mathcal{P}$ for every $g \\in G$ and every part $B \\in \\mathcal{P}$.\n\n## Remarks\n\nEach part of a block system is a block.\nConversely, if $G$ is [[transitive|02GK6HZYBT]] and $B$ is a block, then $\\set{gB : g \\in G}$ is a block system.\nFor transitive $G$ all parts of a block system have the same size, so for finite $\\Omega$ this size divides $\\abs{\\Omega}$.\nThe partition into singletons and the partition $\\set{\\Omega}$ are the *trivial* block systems.\nA transitive action whose only block systems are trivial is [[primitive|Y10S0751KG]].\n\nExample: the eight symmetries of a square acting on the corners $1, 2, 3, 4$ in cyclic order, and $\\mathcal{P} = \\set{\\set{1, 3}, \\set{2, 4}}$.\nEach symmetry maps a pair of opposite corners to a pair of opposite corners, so it maps each part of $\\mathcal{P}$ to a part of $\\mathcal{P}$.\n\nNon-example: for the same action, $\\mathcal{P}' = \\set{\\set{1, 2}, \\set{3, 4}}$.\nThe quarter turn $(1\\,2\\,3\\,4)$ maps $\\set{1, 2}$ to $\\set{2, 3}$, which is not a part of $\\mathcal{P}'$.\n\nIf $G$ is transitive and $\\omega \\in \\Omega$, the blocks containing $\\omega$ correspond to the subgroups $H$ with $G_\\omega \\le H \\le G$, where $G_\\omega$ is the [[stabiliser|0AYMP41THN]]: $H$ gives the block $H\\omega$.\nThe [[orbits|0YQPZB5Q1G]] of a [[normal subgroup|0CB0DX5BVV]] of $G$ form a block system.\n",
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