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  "body": "\n## Summary\n\nAn amenable group is a group with a finitely additive, translation-invariant way to measure the size of all of its subsets, with total size $1$.\nFinite groups, abelian groups and solvable groups are amenable; groups containing a non-abelian [[free group|0KTW26C80J]] are not.\nAmenability has many equivalent forms, such as the Følner condition.\nIt governs the existence of paradoxical decompositions such as the one in the [[Banach–Tarski paradox|FH250K068Z]].\n\n## Explanations\n\n### L1 human\n\nTry to assign to every subset of a group the \"fraction of the group\" it fills.\nDisjoint pieces should add up, and moving a set by a group element should not change its fraction.\nFor a finite group this is easy: take the number of elements divided by the order of the group.\nFor the integers one can do it too, with a kind of limiting density, though not by an explicit formula.\nFor the free group on two generators it is impossible: the group can be cut into pieces that, after moving, make two full copies of the group.\nAmenable groups are those where such a fair notion of size exists.\n",
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