An Elementary Proof of the Nonexistence of Tarski Monster Groups of Exponent 3
Hiroshi Arai
TL;DR
The paper addresses the nonexistence of Tarski monster groups of exponent $3$, a fact usually established via deep results in the restricted Burnside problem and Burnside-type identities. It presents an fully elementary argument centered on the key equation $aa^g = a^g a$ that holds in every group of exponent $3$, with a remark connecting it to Hall's work on the free Burnside group. It then proves that any infinite group of exponent $3$ contains an infinite abelian subgroup by a case analysis on conjugacy classes, leading to a contradiction with the Tarski monster hypothesis. Consequently, no Tarski monster group of exponent $3$ exists, and the method provides an accessible, self-contained proof avoiding heavy prerequisites. This contributes to the educational toolkit for group theory by isolating a single simple identity as the core of the argument.
Abstract
It is well known that Tarski monster groups of exponent~3 do not exist. Traditional proofs rely on deep structural results, such as the restricted Burnside problem, properties of the free Burnside group, or Engel-type identities, and involve substantial technical computations. In this note we give a fully elementary proof: the argument reduces to a single simple identity, expressed as a key equation, whose verification requires only a brief calculation.