Madhya of a Multiset Group and Its Structural Properties

Multiset theory provides an effective generalization of classical set theory by allowing repeated occurrences of elements through frequency functions. When multisets are defined over algebraic structures such as groups, rich and meaningful extensions arise. In this paper, a new concept called Madhya of an Mgroup is introduced and studied in detail. The notion is motivated by the ideas of abelian groups and abelian Mgroups, where commutativity plays a central role. The Madhya of an Mgroup is defined using symmetry conditions on frequency functions, and several illustrative examples are discussed over well-known groups such as the quaternion group, power set groups under symmetric difference, and permutation groups. Important properties of Madhya are investigated, including its relationship with the original Mgroup. A series of theorems are proved to establish that Madhya forms a meaningful and structurally significant sub-multiset of an Mgroup. These results strengthen the link between multiset theory and group theory and open new directions for further research in multiset-based algebraic structures.