---
tag: 5RFPT46DXX
kind: statement
role: theorem
name: Oates–Powell theorem
msc: ["20E10", 08B05]
uses: [YGYZAQ12MZ, G5S1M9, PKD440G2XW]
about: [YGYZAQ12MZ]
sources:
  - url: https://en.wikipedia.org/wiki/Variety_(universal_algebra)
---

## Statement

Let $G$ be a finite [[group|G5S1M9]].
Then $G$ is [[finitely based|YGYZAQ12MZ]]: there is a finite set of [[identities|PKD440G2XW]] satisfied by $G$ from which every identity of $G$ follows.

## 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.

## Explanations

### L1 human

The laws satisfied by a finite group can always be summarised by a finite list.
For example, the laws of a finite abelian group of exponent $n$ follow from the group axioms, commutativity and $x^n \approx e$.
The theorem says that every finite group, abelian or not, has such a finite list of laws.

### L2 human

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.