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  "body": "\n## Statement\n\nLet $G$ be a finite [[group|G5S1M9]].\nThen $G$ is [[finitely based|YGYZAQ12MZ]]: there is a finite set of [[identities|PKD440G2XW]] satisfied by $G$ from which every identity of $G$ follows.\n\n## Remarks\n\nOates and Powell proved that the variety generated by a finite group is a Cross variety.\nThis means that it is finitely based, it is locally finite, and it contains only finitely many critical groups.\nA finite group is critical if it does not lie in the variety generated by its proper subgroups and proper quotients.\nThe proof is a long induction that controls the finite groups in the variety through these critical groups.\n\nExample: the symmetric group $S_3$ is finitely based, although its equational theory is not as simple as that of abelian groups.\n\nThe group axioms matter: a finite semigroup need not be finitely based.\nPerkins proved that the six-element Brandt monoid $B_2^1$ is not finitely based as a semigroup.\nThe analogous statement for finite rings holds, by theorems of Kruse and of L'vov.\n\n## Explanations\n\n### L1 human\n\nThe laws satisfied by a finite group can always be summarised by a finite list.\nFor example, the laws of a finite abelian group of exponent $n$ follow from the group axioms, commutativity and $x^n \\approx e$.\nThe theorem says that every finite group, abelian or not, has such a finite list of laws.\n\n### L2 human\n\nThe theorem answers the finite basis question for groups, a special case of Tarski's finite basis problem for finite algebras.\nIt is the group case of a family of results: finite rings and finite lattices are also finitely based, while finite semigroups and finite groupoids may fail to be.\nThe notion of a Cross variety from the proof is a main tool in the theory of varieties of groups.\nThe result is stated with the group signature of multiplication, inverse and identity, so the identities are group words.\n",
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  "msc": [
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  "name": "Oates–Powell theorem",
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