Scotland U.K.

The search for single axioms for groups has long interested mathematicians. In 1938, Tarski [7] presented the following single equational axiom (in terms of subtraction) for Abelian groups: x (y (z (x y))) = z; (1) and in 1952, Higman and Neumann [1] presented the following single equational axiom (in terms of division) for ordinary groups: (x=((((x=x)=y)=z)=(((x=x)=x)=z))) = y: (2) We use additive notation, +, 0, 0, , for Abelian groups, and multiplicative notation, , e, 1, =, for ordinary groups. Throughout this note, and = are binary operations rather than abbreviations for, e.g., x+ y 0 and x