Jordan-Hölder Theorem for Multigroups

Abstract

Multigroup theory is the application of multisets to the theory of groups. Many group's theoretic notions have been studied in multigroup theory, however, the ideas of maximal normal subgroup, simple group, normal series, composition series, and the Jordan-Holder Theorem are yet to be investigated in multiset context. In this article, we define simple multigroup, maximal normal submultigroup, normal series for multigroup, and composition series for multigroup with examples. With these concepts, we establish the Jordan-Holder Theorem in multigroup theory. It is shown that every finite multigroup defined over a finite group has a composition series. In addition, it is established that every finite multigroup defined over a finite group has at least two composition series which are equivalent.