**Authors:** N. C. Fiala and K. M. Agre

### Abstract

We study identities in product and a constant e only that are valid in all groups of exponent 2 (3) with neutral element e and that imply that a groupoid satisfying one of them is a group of exponent 2 (3) with neutral element e. Such an identity will be called a single axiom with neutral element for groups of exponent 2 (3). We utilize the automated reasoning software Prover9 and Mace4 to attempt to find all shortest single axioms with neutral element for groups of exponent 2 (3). Beginning with a list of 1323 (1716) candidate identities that contains all shortest possible single axioms with neutral element for groups of exponent 2 (3), we find 173 (148) single axioms with neutral element for groups of exponent (2) 3 and eliminate all but 5 (119) of the remaining identities as not being single axioms with neutral element for groups of exponent 3. We also prove that a finite model of any of these 5 (119) identities must be a group of exponent 2 (3) with neutral element e.

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