On the µF-subgroups of Some Finite Abelian Groups

The paper presented here introduces and explores the concept of the subgroup determined by Möbius function, denoted as the µF-subgroup, within the context of finite cyclic groups C n. It makes significant contributions to the field of group theory by investigating the properties and relationships of these µF- subgroups within different group structures. One of the primary findings of this paper is the assertion that within finite cyclic groups C n , the collection of all µF-subgroups, denoted as LµF (C n ), forms a sub lattice of the lattice L (C n ). This result is notable because it establishes a specific structure within the lattice of subgroups of cyclic groups. Furthermore, the paper identifies a fundamental connection between Hall subgroups and µF-subgroups, emphasizing that every Hall subgroup of a group qualifies as a µF-subgroup. This connection sheds light on the broader relevance and significance of µF-subgroups in group theory. The paper extends its investigation to the product of cyclic groups, C m × C n , and explores the meet and join operations of subgroups within this product group. It proves that the lattice of µF- subgroups, denoted as LµF (C m × C n ), is not necessarily a sub lattice of the lattice L (C m × C n ). However, the paper provides the condition when LµF (C m × C n ) forms a lattice, and the methods to determine the meet and join of any two µF-subgroups within this context. A significant contribution of the paper lies in establishing a characterization for LµF (C m ×C n ) to be a sub lattice of L (C m × C n ) and specifying the conditions under which this occurs. This characterization adds depth to our understanding of when and how sub lattices can be formed within the lattice of subgroups in a product group. Finally, the paper explores the cardinality of the set LµF (C m × C n ) for various values of m and n, providing insights into the size and complexity of these µF-subgroups within the product group.