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Vipul Kakkar

Publications and source records attributed to Vipul Kakkar.

At least 19 recordsLinked to original sources
arXiv2026

Commutator identities, Lie product identities, and Multiplicative Lie algebras

The aim of this paper is to give a formal meaning of universal commutator identities and to see as to how the structure of multiplicative Lie algebra on a group is a potential candidate to describe universal commutator identities. We introduce some higher Schur multipliers. We shall also introduce and study Steinberg and Chevalley Multiplicative Lie algebras, the multiplicative Lie algebras with commutator identities among the root vectors describing Lie product relations. Free multiplicative Lie algebra on Steinberg and Chevalley groups are discussed.

math.GR↗
arXiv2026

Impartial games on two finite Groups

In this paper, we study impartial achievement games and impartial avoidance games introduced by Anderson and Harary. Using the criteria of maximal subgroups, we study the game for Frobenius groups and non-abelian groups with all abelian subgroups.

math.GR↗
arXiv2025

Determinant for automorphisms of semidirect product of groups

A description of the endomorphisms of semidirect products of two groups as a group of $2\times 2$ matrices of maps is already known. Using this description, we have studied the concept of determinant for the endomorphisms of semidirect product of two groups. A characterization of the invertible endomorphisms is given with the help of the tools developed using the determinants.

math.GR↗
arXiv2025

Line Graph Characterization of Cyclic Subgroup Graph

The cyclic subgroup graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with cyclic subgroups as a vertex set and two distinct vertices $H_1$ and $H_2$ are adjacent if and only if $H_1 \leq H_2$ and there does not exist any cyclic subgroup $K$ such that $H_1 < K < H_2$. In this paper, we classify all the finite groups $G$ such that $Γ(G)$ is the line graph of some graph.

math.CO↗
arXiv2024

The Minimal (Edge) Connectivity of Some Graphs of Finite Groups

In this paper, we classify all the finite groups $G$ such that the commuting graph $Γ_C(G)$, order-sum graph $Γ_{OS}(G)$ and non-inverse graph $Γ_{NI}(G)$ are minimally edge connected graphs. We also classify all the finite groups $G$ for that, these graphs are minimally connected. We also classify some groups for that the co-prime graph $Γ_{CP}(G)$ has minimal edge connectedness. In final part, we classify all the finite groups $G$ for that co-prime graph $Γ_{CP}(G)$ is minimally connected.

math.CO↗
arXiv2024

Finite groups whose commuting graphs are line graphs

The commuting graph ${Γ(G)}$ of a group $G$ is the simple undirected graph with group elements as a vertex set and two elements $x$ and $y$ are adjacent if and only if $xy=yx$ in $G$. By eliminating the identity element of $G$ and all the dominant vertices of $Γ(G)$, the resulting subgraphs of $Γ(G)$ are $Γ^*(G)$ and $Γ^{**}(G)$, respectively. In this paper, we classify all the finite groups $G$ such that the graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the line graph of some graph. We also classify all the finite groups $G$ whose graph $Δ(G) \in \{Γ(G), Γ^*(G), Γ^{**}(G)\}$ is the complement of line graph.

math.CO↗
arXiv2024

Associating hypergraphs defined on loops

In this paper, we define a new hypergraph $\mathcal{H(V,E)}$ on a loop $L$, where $\mathcal{V}$ is the set of points of the loop $L$ and $\mathcal{E}$ is the set of hyperedges $e=\{x,y,z\}$ such that $x,y$ and $z$ associate in the order they are written. We call this hypergraph as the associating hypergraph on a loop $L$. We study certain properites of associating hypergraphs on the Moufang loop $M(D_n,2)$, where $D_n$ denotes the dihedral group of order $2n$.

math.CO↗
arXiv2024

Semidirect Product of Loops with Groups

In this paper, we have studied the loops which are the semidirect products of a loop and a group. Also, the cummutant, nuclei and the center of such loops are studied.

math.GR↗
arXiv2023

Gyro-Groups, Gyro-splittings and Co-homology

In this paper, we study gyro-groups associated to groups, group extensions admitting gyro-sections, and corresponding co-homologies. We also describe the obstructions in terms of co-homomology. The notion of gyro-Schur Multiplier and that of gyro-Milnor $K_{2}$ group are introduced.

math.GR↗
arXiv2021

Automorphisms of Zappa-Szép Product

In this paper, we have found the automorphism group of the Zappa-Szép product of two groups. Also, we have computed the automorphism group of the Zappa-Szép product of a cyclic group of order $m$ and a cyclic group of order $p^{2}$, where $p$ is a prime.

math.GR↗
arXiv2021

Automorphisms of semidirect products fixing the non-normal subgroup

In this paper, we describe the automorphism group of semidirect product of two groups that fixes the non-normal subgroup of it. We have computed these automorphisms for the non-abelian metacyclic $p$-group and non-abelian $p$-groups $(p\ge 5)$ of order $p^{4}$, where $p$ is a prime.

math.GR↗
arXiv2021

Gyrogroups associated with groups

In this paper, we study the properties of the associated gyrogroup ${^\circ}G$ of a given group $G$ of nilpotency class $3$. We have proved that if $3$ does not divide the order of the group $G$, then the nilpotency class of the associated gyrogroup ${^\circ}G$ is same as that of the group $G$. We have also studied the problem of abelian inner mapping group in this context.

math.GR↗
arXiv2018

A Construction of Bent Functions on a finite group

In this paper, we discuss when a class function on a finite group is a bent function. We have found a necessary condition for a class function on a finite abelian group to be bent. Also, we have found a necessary and sufficient condition for a class function on a finite cyclic group to be bent.

math.CO↗
arXiv2018

On Commuting Graphs of Generalized Dihedral Groups

For a group $G$ and a subset $X$ of $G$, the commuting graph of $X$, denoted by $Γ(G,X)$ is the graph whose vertex set is $X$ and any two vertices $u$ and $v$ in $X$ are adjacent if and only if they commute in $G$. In this article, certain properties of the commuting graph of generalized dihedral groups have been studied.

math.CO↗
arXiv2018

The Schur Multipliers of $p$-Groups of Order $p^5$

In this article, we compute the Schur multiplier, non-abelian tensor square and exterior square of non-abelian $p$-groups of order $p^5$. As an application we determine the capability of groups of order $p^5$.

math.GR↗
arXiv2015

A Study of Groups through Transversals

In this note, a necessary and sufficient condition for the normalizer of a core-free subgroup $H$ of a finite group $G$ to be normal in $G$ is obtained. Also, a known result of finite groups is obtained through transversal.

math.GR↗