부산대학교 도서관

Extensions of partial endomorphisms of groups have been studied in the past by \n B. H. Neumann\en and \n H. Neumann\en [Proc. London Math. Soc. (3) {\bf 2} (1952), 337--348; MR0050586 (14,351b)], by Chehata [Proc. Glasgow Math. Assoc. {\bf 2} (1954), 37--46; MR0062742 (16,10e); Canad. J. Math. {\bf 15} (1963), 766--770; MR0154908 (27 \#4852)] and by Chehata and Sherif [Math. Sci. {\bf 4} (1979), no. 2, 99--104; MR0561997 (81c:20027)]. The problem is to find conditions under which a partial endomorphism [resp. a set of partial endomorphisms] of a group $G$, i.e., a homomorphism of a subgroup $A$ of $G$ onto a subgroup $B$ of $G$, can be extended [resp. can be simultaneously extended] to a ``total'' endomorphism of a group $G^*$ containing $G$ as a subgroup. In the present paper the authors extend all the above results to $M$-groups, i.e., groups with a set of operators $M$, in the usual sense.