Oates–Powell theorem

Remarks

Oates and Powell proved that the variety generated by a finite group is a Cross variety. This means that it is finitely based, it is locally finite, and it contains only finitely many critical groups. A finite group is critical if it does not lie in the variety generated by its proper subgroups and proper quotients. The proof is a long induction that controls the finite groups in the variety through these critical groups.

Example: the symmetric group $S_3$ is finitely based, although its equational theory is not as simple as that of abelian groups.

The group axioms matter: a finite semigroup need not be finitely based. Perkins proved that the six-element Brandt monoid $B_2^1$ is not finitely based as a semigroup. The analogous statement for finite rings holds, by theorems of Kruse and of L'vov.

Going deeper

For a student

The theorem answers the finite basis question for groups, a special case of Tarski's finite basis problem for finite algebras. It 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. The notion of a Cross variety from the proof is a main tool in the theory of varieties of groups. The result is stated with the group signature of multiplication, inverse and identity, so the identities are group words.