Searcharxiv⌕ Search

arXiv subjects

Sugandha Maheshwary

Publications and source records attributed to Sugandha Maheshwary.

18 recordsLinked to original sources
arXiv2026

Prime and N-Prime graphs of solvable uniformly semi-rational groups

In this paper, we study the prime graph, also known as Gruenberg-Kegel graph, realizable by finite solvable uniformly semi-rational groups. This is primarily achieved by classifying the prime graphs realizable by metanilpotent groups. We also introduce a new class of groups, called MP* groups, which extends the class of groups with Magnus property and forms a natural subclass of uniformly semi-rational groups. For MP* groups, we give a complete classification of the prime graphs realized by them. In the process, we obtain the realizability of prime graphs by uniformly semi-rational groups, assuming them to be in varied substantial subclasses of solvable groups, namely, 2-Frobenius, metacyclic, metabelian, nilpotent-by-abelian, abelian-by-cyclic and cyclic-by-abelian. We also classify the so called N-prime graphs for all these classes of groups. The prime graph question for uniformly semi-rational groups has also been addressed.

math.GR↗
arXiv2026

A Study Of Skew-Polycyclic Codes Over A Non-Chain Ring

For a prime \(p\) and a positive integer \(m\), let \(\mathbb{F}_{p^m}\) be the finite field of cardinality \(p^m\), and let $ R_{u^2,v^2,p^m} =\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}+v\mathbb{F}_{p^m} +uv\mathbb{F}_{p^m}, ~ u^2=v^2=0,\ uv=vu, $ be a finite non-chain ring. In this paper, we study skew polycyclic codes of length \(lj\) associated with \(f(x)^j\), where \(f(x)\) is a central polynomial of degree \(l\) in $R_{u^2, v^2, p^m}[x; Θ],$ where $Θ$ being an automorphism of \(R_{u^2,v^2,p^m}\). We describe these codes, characterize free skew polycyclic codes, and determine their ranks. Under suitable centrality assumptions, we decompose the quotient ring associated with \(x^{np^s}-λ\), where \(\gcd(n,p)=1\) and \(Θ(λ)=λ\). This reduces the study of skew \((λ,Θ)\)-constacyclic codes of length \(np^s\) to the study of left ideals of $\frac{R_{u^2,v^2,p^m}[x;Θ]}{\langle f(x)^j\rangle}, $ where \(f(x)\) is a central irreducible divisor of degree \(l\) of \(x^{np^s}-λ\), for an invertible element \(λ\in R_{u^2,v^2,p^m}\) and \(j\in\mathbb{N}\). We then apply these results to skew \((λ,Θ)\)-constacyclic codes of length \(p^s\) for different classes of units \(λ\). Several examples are presented to illustrate the theory and to obtain optimal codes. Finally, when \(Θ\) is the identity automorphism, we study constacyclic codes of length \(np^s\) over \(R_{u^2,v^2,p^m}\), according as \(x^n-α_0\) is irreducible or reducible over \(\mathbb{F}_{p^m}\). These results extend the work of \cite{CCDF18} and \cite{ZTG18} on constacyclic codes of length \(np^s\) over \(\mathbb{F}_{p^m}+u\mathbb{F}_{p^m}\) to the finite non-chain ring \(R_{u^2,v^2,p^m}\).

cs.IT↗
arXiv2026

Skew Constacyclic Codes Of Length $np^s$ over $ \frac{\mathbb{F}_{p^m}[u]}{\langle u^k \rangle}

Let $\mathbb{F}_{p^m}$ be the field containing $p^m$ elements where $p$ is an odd prime and $m \in \mathbb{N}$. In this article, we propose a unified approach to the study of skew constacyclic codes of length $np^s$ over the ring $R_k = \mathbb{F}_{p^m}[u]/\langle u^k \rangle,$ where $n, s, k \in \mathbb{N}$ and $\gcd(n, p)=1$. Consider the skew polynomial ring $R_k[x;Θ]$, where $Θ$ is an automorphism of $R_k$ such that $xa = Θ(a)x$ for all $a \in R_k$. Let $f(x)$ be a central irreducible divisor of $x^{np^s} - λ$ of degree $l$ and multiplicity $j$ in $R_k[x;Θ]$, where $λ$ is an invertible element in $R_k$. In this article, we study skew constacyclic codes of length \(np^s\) over \(R_k\), which reduces to the study of skew polycyclic codes of length $jl$ associated with a polynomial \(f(x)^j\). Using the fact that skew polycyclic codes associated with a polynomial \(f(x)^j\) can be described by the left ideal structure of the quotient ring $R_k[x;Θ]/\langle f(x)^{j}\rangle$, we investigate this class of codes for specific choices of $Θ$. In particular, if $λ$ is an invertible element of $\mathbb{F}_{p^m}$, we classify all left ideals and establish an isomorphism between skew cyclic and skew constacyclic codes, under suitable conditions. Furthermore, we provide a comprehensive analysis of skew constacyclic codes of length $3p^s$ over $R_k$. Finally, we examine skew cyclic and skew negacyclic codes of length $6p^s$ over $R_k$ using the factorization of $x^{6p^s} - 1$ and $x^{6p^s} + 1$, respectively; with a complete case-by-case analysis. Examples demonstrating codes with optimal parameters are also included.

cs.IT↗
arXiv2026

Gruenberg-Kegel graphs: cut groups, rational groups and the Prime Graph Question

The Gruenberg-Kegel graph of a group is the undirected graph whose vertices are those primes which occur as the order of an element of the group, and distinct vertices $p$, $q$ are joined by an edge whenever the group has an element of order $pq$. It reflects interesting properties of the group. A group is said to be cut if the central units of its integral group ring are trivial. This is a rich family of groups, which contains the well studied class of rational groups, and has received attention recently. In the first part of this paper we give a complete classification of the Gruenberg-Kegel graphs of finite solvable cut groups which have at most three elements in their prime spectrum. For the remaining cases of finite solvable cut groups, we strongly restrict the list of the possible Gruenberg-Kegel graphs and realize most of them by finite solvable cut groups. Likewise, we give a list of the possible Gruenberg-Kegel graphs of finite solvable rational groups and realize as such all but one of them. As an application, we completely classify the Gruenberg-Kegel graphs of metacyclic, metabelian, supersolvable, metanilpotent and $2$-Frobenius groups for the classes of cut groups and rational groups, respectively. The Prime Graph Question asks whether the Gruenberg-Kegel graph of a group coincides with that of the group of normalized units of its integral group ring. The recent appearance of a counter-example for the First Zassenhaus Conjecture on the torsion units of integral group rings has highlighted the relevance of this question. We answer the Prime Graph Question for integral group rings for finite rational groups and most finite cut groups.

math.GR↗
arXiv2026

The structure of $\mathbb{Z}_nG$ and its unit group

This article determines the structure of the group ring $\mathbb{Z}_nG$, where $G$ is a finite group and $\mathbb{Z}_n$ is the ring of integers modulo $n$, such that $n$ is relatively prime to the order of $G$. The decomposition of $\mathbb{Z}_nG$ is given as a direct sum of matrix rings over Galois rings, thereby extending the structural theory of group rings beyond the classical field setting. We also provide a method to compute a generating set of the unit group $\mathcal{U}(\mathbb{Z}_nG)$, in terms of elementary matrices, using Shoda pair theory. The results are illustrated with examples.

math.RA↗
arXiv2025

On metacyclic p-group codes

In this article, we study the metacyclic p-group codes arising from finite semisimple group algebras. In [CM25], we studied group codes arising from metacyclic groups with order divisible by two distinct odd primes. In the current work, we focus on metacyclic p-group codes, as a result of which we are also able to extend the results of [CM25] for metacyclic groups with order divisible by any two primes, not necessarily odd or distinct. Consequently, existing results on group algebras of some important classes of groups, including dihedral and quaternion groups, have been extended. Additionally, we provide left codes for the undertaken group algebras. Finally, we construct non-central codes using units motivated by Bass and bicyclic units, which are inequivalent to any abelian group codes and yield best known parameters.

math.RA↗
arXiv2025

Cut Groups: The Progress And The Problems

This article focuses on the study of cut groups, i.e., the groups which have only trivial central units in their integral group ring. We provide state of art for cut groups. The results are compiled in a systematic manner and have also been analogously studied for some generalised classes, such as quadratic rational groups and semi-rational groups etc. For these bigger classes, some results have been extended, while others have been posed as questions. The progress and the problems signify the development and the potential that holds in the topic of cut groups and its generalisations.

math.RA↗
arXiv2025

Extended cut groups

A finite group G is said to be a cut group if all central units in the integral group ring ZG are trivial. In this article, we extend the notion of cut groups, by introducing extended cut groups. We study the properties of extended cut groups analogous to those known for cut groups and also characterise some substantial classes of groups having the property of being extended cut. A complete classification of extended cut split metacyclic groups has been presented.

math.GR↗
arXiv2021

Global and local properties of finite groups with only finitely many central units in their integral group ring

The aim of this article is to explore global and local properties of finite groups whose integral group rings have only trivial central units, so-called cut groups. For such a group we study actions of Galois groups on its character table and show that the natural actions on the rows and columns are essentially the same, in particular the number of rational-valued irreducible characters coincides with the number of rational-valued conjugacy classes. Further, we prove a natural criterion for nilpotent groups of class 2 to be cut and give a complete list of simple cut groups. Also, the impact of the cut property on Sylow 3-subgroups is discussed. We also collect substantial data on groups which indicates that the class of cut groups is surprisingly large. Several open problems are included.

math.GR↗
arXiv2021

Abelianization of the unit group of an integral group ring

For a finite group $G$ and $U: = U(\mathbb{Z}G)$, the group of units of the integral group ring of $G$, we study the implications of the structure of $G$ on the abelianization $U/U'$ of $U$. We pose questions on the connections between the exponent of $G/G'$ and the exponent of $U/U'$ as well as between the ranks of the torsion-free parts of $Z(U)$, the center of $U$, and $U/U'$. We show that the units originating from known generic constructions of units in $\mathbb{Z}G$ are well-behaved under the projection from $U$ to $U/U'$ and that our questions have a positive answer for many examples. We then exhibit an explicit example which shows that the general statement on the torsion-free part does not hold, which also answers questions from [BJJ$^+$18].

math.RA↗
arXiv2020

Units and Augmentation Powers in Integral Group Rings

The augmentation powers in an integral group ring $\mathbb ZG$ induce a natural filtration of the unit group of $\mathbb ZG$ analogous to the filtration of the group $G$ given by its dimension series $\{D_n(G)\}_{n\ge 1}$. The purpose of the present article is to investigate this filtration, in particular, the triviality of its intersection.

math.RA↗
arXiv2018

Group rings and the RS-property

The object of this paper is to study (infinite) groups whose integral group rings have only trivial central units. This property is closely related to a property, here called the RS-property (\cite{DMS05}, \cite{RS90}), involving conjugacy in the group.

math.RA↗
arXiv2017

Finite semisimple group algebra of a normally monomial group

In this paper, the complete algebraic structure of finite semisimple group algebra of a normally monomial group is described. The main result is illustrated by computing the explicit Wedderburn decomposition of finite semisimple group algebras of various normally monomial groups. The automorphism groups of these group algebras are also determined.

math.RA↗
arXiv2016

Integral group rings with all central units trivial

The object of study in this paper is the finite groups whose integral group rings have only trivial central units. Prime-power groups and metacyclic groups with this property are characterized. Metacyclic groups are classified according to the central height of the unit groups of their integral group rings.

math.RA↗
arXiv2015

Extremely strong Shoda pairs with \texttt{GAP}

We provide algorithms to compute a complete irredundant set of extremely strong Shoda pairs of a finite group $G$ and the set of the primitive central idempotents of the rational group algebra $\mathbb{Q}[G]$ realized by them. These algorithms are also extended to write new algorithms for computing a complete irredundant set of strong Shoda pairs of $G$ and the set of the primitive central idempotents of $\mathbb{Q}[G]$ realized by them. Another algorithm to check whether a finite group $G$ is normally monomial or not is also described.

math.RA↗
arXiv2014

On the index of a free abelian subgroup in the group of central units of an integral group ring

Let $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ denote the group of central units in the integral group ring $\mathbb{Z}[G]$ of a finite group $G$. A bound on the index of the subgroup generated by a virtual basis in $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ is computed for a class of strongly monomial groups. The result is illustrated with application to the groups of order $p^{n}$, $p$ prime, $n \leq 4$. The rank of $\mathcal{Z}(\mathcal{U}(\mathbb{Z}[G]))$ and the Wedderburn decomposition of the rational group algebra of these $p$-groups have also been obtained.

math.RA↗