On the group properties of substitutions defined on the subgroup cosets of finite Abelian group of composite order

Necessary and sufficient conditions of transitivity and primitivity of substitution groups generated by substitutions on the cosets of the Abelian group $G$ are obtained. The structure of these groups is described for the case of their primitivity and for the case when $G$ is the direct product of its subgroups. Necessary and sufficient conditions of primitivity of the group generated by the right regular representation of the group $G$ and a substitution defined on the cosets of the group $G$ are obtained also.