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Vivek Kumar Jain

Publications and source records attributed to Vivek Kumar Jain.

8 recordsLinked to original sources
arXiv2026

On the number of orthomorphisms of the alternating group on four symbols

In this study, the set of orthomorphisms of a finite group G of order 3M with a normal subgroup of order M has been partitioned into classes. Additionally, a number of relationships between the orders of these classes have been established. Ultimately, this approach has been used to theoretically determine all 3840 orthomorphisms of alternating groups on four symbols by just counting 30 orthomorphisms, a problem that had been unsolved since 1992.

math.GR↗
arXiv2024

A Characterization of Group Through Isomorphism Classes of Transversals

Let G be a group and H a subgroup of G of finite index. In this article, it is proved that if the number of isomorphism classes of right transversals of H in G is 5, then the index of H in G is 6 and the permutation representation of G on right cosets of H in G is isomorphic to the alternating group on four symbols.

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↗
arXiv2014

Solvable and Nilpotent Right Loops

In this paper the notion of nilpotent right transversal and solvable right transversal has been defined. Further, it is proved that if a core-free subgroup has a generating solvable transversal or a generating nilpotent transversal, then the whole group is solvable.

math.GR↗
arXiv2013

A Note On Transversals

Let $G$ be a finite group and $H$ a core-free subgroup of $G$. We will show that if there exists a solvable, generating transversal of $H$ in $G$, then $G$ is a solvable group. Further, if $S$ is a generating transversal of $H$ in $G$ and $S$ has order 2 invariant sub right loop $T$ such that the quotient $S/T$ is a group. Then $H$ is an elementary abelian 2-group.

math.GR↗
arXiv2012

On The Isomorphism Classes Of Transversals III

Let $G$ be a finite group and $H$ a subgroup of $G$. Each left transversal (with identity) of $H$ in $G$ has a left loop (left quasigroup with identity) structure induced by the binary operation of $G$. We say two left transversals are isomorphic if they are isomorphic with respect to the induced left loop structures. In this paper, we develop a method to calculate the number of isomorphism classes of transversals of $H$ in $G$. Also with the help of this we calculate the number of non-isomorphic left loops of a given order.

math.GR↗
arXiv2009

On Odd Order Nilpotent Groups With Class 2

Let $G$ be an odd order nilpotent group with class 2 and $e$ denotes the exponent of its commutator subgroup. Let $e=p_1^{r_1}p_2^{r_2}... p_s^{r_s}$, where $p_i$'s are odd primes and $r_i$'s are non-negative integers. Then there are at least $r_1+r_2+... +r_s$ non-isomorphic nilpotent groups with class two and the order of each of the group is equal to the order of $G$.

math.GR↗