A groupoid A= ⟨A;∗⟩ is
called a left (resp. right) difference group if there is a binary operation +
in A such that the system ⟨A;+⟩ is an abelian group and x ∗ y = −x + y
(resp. x ∗ y = x − y). A symmetric difference group is a groupoid satisfying all the
identities common to both left and right difference groups. In this note we determine
the structure of a symmetric difference group. Using this, we show that any finitely
based equational theory of symmetric difference groups is one-based. This
includes the known result that the theories of left and right difference groups
are one-based. Other known results on finitely based theories of rings also
follow.
