On the µF-subgroups of Some Finite Abelian Groups


Shailesh Singh
Department of Mathematics, Savitribai Phule Pune University
Vilas Kharat
Department of Mathematics, Savitribai Phule Pune University
Manish Agalave
Department of Mathematics, Savitribai Phule Pune University


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.

October 19, 2023