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  "body": "\n## Summary\n\nA Frobenius group is a finite [[transitive|02GK6HZYBT]] permutation group in which some non-identity element fixes a point, but no non-identity element fixes two points.\nThe elements that fix no point, together with the identity, form a normal subgroup, the Frobenius kernel, by [[Frobenius's theorem|XK7H3YXBQT]].\nThe group is then a [[semidirect product|0H3WZXCZNF]] of the kernel and a point [[stabiliser|0AYMP41THN]], the Frobenius complement.\nFrobenius groups appear throughout finite group theory and character theory.\n\n## Explanations\n\n### L1 human\n\nTake the three corners of a triangle and all six ways to permute them.\nA reflection keeps one corner in place and swaps the other two; a rotation moves every corner.\nNo symmetry other than the identity keeps two corners in place.\nThis is the smallest Frobenius group.\nIts rotations, together with the identity, form a subgroup, and that is the pattern in every Frobenius group.\n",
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