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  "body": "\n## Summary\n\nA permutation group is primitive when it is [[transitive|02GK6HZYBT]] and preserves no nontrivial [[block system|SF738NKWJ0]], that is, no nontrivial splitting of the set into parts.\nPrimitive groups are the building blocks of all finite permutation groups.\nA transitive group of degree $n$ that is not primitive embeds in a [[wreath product|W1X984PKJ9]] $S_k \\wr S_m$ with $km = n$ and $1 \u003c k, m \u003c n$.\nA transitive group is primitive exactly when its point [[stabilisers|0AYMP41THN]] are maximal subgroups.\nThe [[O'Nan–Scott theorem|Z4WVANZPK4]] sorts the finite primitive groups into a few types.\n\n## Explanations\n\n### L1 human\n\nImagine a group that shuffles a set of points.\nIf the group always keeps some grouping of the points together, for example pairs of opposite corners of a square, then it is not primitive.\nA primitive group has no such hidden grouping: the only groupings it respects are the trivial ones, all points in one part or each point alone.\nThe full group of all permutations of a set is primitive.\nThe symmetries of a square, acting on its corners, are not primitive, because they keep the two diagonals as pairs.\n",
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