arXiv subjects

Sachin Ballal

Publications and source records attributed to Sachin Ballal.

5 recordsLinked to original sources

arXiv2026

Co-maximal Hypergraph on Dn

Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. \textit{The co-maximal hypergraph of $G$}, denoted by $Co_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H K = G \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that the product of any two vertices is equal to $G$. The aim of this paper is to study the co-maximal hypergraph of dihedral groups, $Co_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $Co_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertrees, star structures and 3-uniform hypergraphs of $Co_\mathcal{H}(D_n)$. Further, we discuss the possibilities of $Co_\mathcal{H}(D_n)$ which can be embedded on the plane, torus and projective plane.

math.CO↗

arXiv2026

On Posets of Classes of Subgroups with Same Set of Orders of Elements

In this paper, we study the posets of classes of subgroups of finite group having same set of orders of elements. We show that this poset is a chain only in the case of p-groups and moreover, we characterize all finite groups for which this poset is C2, the chain with two elements. We also show that this poset forms a lattice in the case of finite cyclic and dihedral groups and give a characterization when this lattice is distributive and modular.

math.GR↗

arXiv2025

On Posets of Classes of Automorphic Subgroups of Finite Groups

In [16], Tarnauceanu studied the poset Iso(G), of isomorphic classes of subgroups of a finite group G and proposed several questions for further research. In this paper, we study the poset AutCl(G), of classes of automorphic subgroups of finite group G. We introduce a partial order on AutCl(G) to tackle problem 5 mentioned in §4 of [16]. More precisely, we prove that AutCl(Dn) and AutCl(Q4m) are distributive lattices. Moreover, we characterize all classes of finite groups for which AutCl(G) is a chain.

math.GR↗

arXiv2025

On Intersection and Co-maximal Hypergraph of $\mathbb{Z}_n$

The aim of this paper is to study the intersection hypergraph $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and co-maximal hypergraph $Co_\mathcal{H}(\mathbb{Z}_n)$ on the subgroups of $\mathbb{Z}_n$. We prove that the intersection and co-maximal hypergraph of a finite abelian group are isomorphic. Hence, we focus on $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$ and examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$. Also, we provide characterizations for hypertrees, star structures of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(\mathbb{Z}_n)$.

math.CO↗

arXiv2025

Intersection Hypergraph on D_n

Let $G$ be a group and $S$ be the set of all non-trivial proper subgroups of $G$. The intersection hypergraph of $G$, denoted by $\tildeΓ_\mathcal{H}(G)$, is a hypergraph whose vertex set is $\{H \in S \,\, | \,\, H \cap K = \{e\} \,\, \text{for some} \, K \in S \}$ and hyperedges are the maximal subsets of the vertex set with the property that any two vertices in it have a trivial intersection. The aim of this paper is to study the intersection hypergraph of dihedral groups, $\tildeΓ_\mathcal{H}(D_n)$. We examine some of the structural properties, viz., diameter, girth and chromatic number of $\tildeΓ_\mathcal{H}(D_n)$. Also, we provide characterizations for hypertreees, star structures of $\tildeΓ_\mathcal{H}(D_n)$, and investigate the planarity and non-planarity of $\tildeΓ_\mathcal{H}(D_n)$.

math.CO↗