The Shortest Single Axioms for Groups of Exponent 4

We study equations of the form (= x) which are single axioms for groups of exponent 4, where is a term in product only. Every such must have at least 9 variable occurrences, and there are exactly three such of this size, up to variable renaming and mirroring. These terms were found by an exhaustive search through all terms of this form. Automated techniques were used in two ways: to eliminate many by verifying that (= x) true in some non-group, and to verify that the group axioms do indeed follow from the successful (= x). We also present an improvement on Neumann's scheme for single axioms for varieties of groups. x0. Introduction. If n 1 is an integer, a group of exponent n is a group in which x n is the identity for all elements x. We study equations of the form (= x) which are single axioms for groups of exponent n, where is a term in product only. Note that in our deenition of \exponent n", we do not require that n is the smallest exponent, so, for example, every group of exponent 2 is also a group of exponent 4. The class of groups of exponent \precisely 4" (that is, also satisfying 9y(y 2 6 = e)) cannot be axiomatized by any set of equations. First, some notation on terms. We shall use the binary function symbol t to denote the group product. We shall also sometimes use standard innx algebraic notation as an abbreviation, with products associating to the right. Thus, for example, x y z and xyz both abbreviate the term t(x; t(y; z)). We use exponentiation as a further abbreviation, with x 1 abbreviating x and x n+1 abbreviating x x n. Let RA() result from associating all products in to the right; thus, for example RA(t(t(x; y); t(z; u))) is t(x; t(y; t(z; u))), which is the same as xyzu by our conventions on algebraic notation. Because of the nite exponent, we can express all the group axioms in terms of product only. Thus, we say, a group of exponent n is a model for the following set of three axioms: