ASSOCIATIVE REGULAR OPERATIONS ON GROUPS CORRESPONDING TO A PROPERTY <p

In previous papers [4; 5; 6] we constructed and studied many classes of operations on groups which have the following properties: (1) Associative; (2) Commutative; (3) The resultant group contains isomorphic images of the original factors and is generated by them; (4) In the resultant group the intersection of a factor with normal subgroup generated by the remaining factors is the unit element. Golovin [3] showed how to obtain all operations satisfying the last three conditions and called such operations regular multiplications. Below we construct for every property (P, applicable to groups, a corresponding regular multiplication. If (P persists homomorphically and ceases freely, we shall show that this multiplication is associative. 0~> is said to persist homomorphically if from the fact that a group satisfies (P it follows that every homomorphic image of the group also satisfies (P. On the other hand, (P is said to cease freely if for every set of groups satisfying (P their free product does not satisfy (P. We introduce a number of preliminary concepts which are all contained in Moran [6]. N(X) is said to be a normal subgroup function if it associates with every group G a normal subgroup N(G) of G. However, we are interested in the following more restrictive class of normal subgroup functions. N(X) is said to be a strongly characteristic subgroup function if N(X)) for every homomorphism <p of X. This is equivalent to the condition that the normal subgroup be prescribed in such a manner that when extra defining relations are imposed on the group then the elements which originally belonged to the subgroup continue to remain therein. It seems that most of the well known normal subgroup functions, such as the Frattini subgroup, are of this type. Let F denote the free product of an arbitrary set of groups Ga, aEM, and N(X) be a normal subgroup function, then corresponding to N(X) we have the normal subgroup [N(Ga), GB]F of F. [N(Ga), Gf,]F is the normal subgroup of F generated by all commutators of the form [ga, g$], where gaEN(Ga), gpEGp and a and fi are distinct elements of the index set M. We now define the Ar*-muitiplication of the groups Ga, aEM, by1