Regular products on the set of groups

VIII SUMMARY Generally, algebraic operations defined on the set of groups do not satisfy the associative law. In the second section of this study, a regular product has been introduced and it has been showed that this product is not associative. There is no general method to show that a regular product does not satisfy the associative law. Therefore, different methods have been used in each example in the second section to show that they are nonassociative. The free and the direct products are the regular algebraic operations which have the commutative and associative properties. In this study, a classification of products that is different from these products, but has common properties has been given by using the verbal products. Finally, the verbal products of two groups chosen specifically have been calculated and it has been proved that all regular products of second order cyclic groups are dihedral groups. Furthermore, it has been showed that these products are commutative and associative.