Goursat Lemma on groups with operators

In this paper the usual Goursat lemma, which describes subgroups of the directproduct of two groups and finite number of groups, is generalized to describingsubgroups of two groups with operators and direct product of a finite number ofgroups with operators. Other possible generalizations are discussed and applications characterizing commutative group with operators of the direct product of twogroups with operators are given. Most of these applications are some characterization of commutative sub-group with operators of the direct product of two groupswith operators. Other we give a characterization of subgroups of a direct productwhich are inner semi-direct products. The applications of the Goursat lemma ongroups with operators are also found in the modules. For two modules A and Bon a ring R, we use easily Goursat lemma for groups with operators to characterize the sub-modules of the direct product A × B which are finitely generated,noetherian sub-modules of A × B their divisible and torsion sub-modules. AMS subject classification: 06F07, 18D10