Let J be a finite commutative nilpotent algebra over a field F of characteristic p. J forms an abelian group under the "circle" operation, defined by a o b = a + b + ab β This group is called the quasi regular group of J. Our main purpose is to investigate the relationship between the structure of J as an algebra, and the structure of its quasi regular group. THEOREM 1.1. The quasi regular group of J is isomorphie to G{p, u;s u Proof. Since the pth power oί xeJ with respect to the operation "o" is x