Single axioms for groups and Abelian groups with various operations

This paper summarizes the results of an investigation into single axioms for groups, both ordinary and Abelian, with each of following six sets of operations: fproduct, inverseg, fdivisiong, fdouble division, identityg, fdouble division, inverseg, fdivision, identityg, and fdivision, inverseg. In all but two of the twelve corresponding theories, we present either the rst single axioms known to us or single axioms shorter than those previously known to us. The automated theorem-proving program Otter was used extensively to construct sets of candidate axioms and to search for and nd proofs that given candidate axioms are in fact single axioms.