arXiv subjects
Publications and source records attributed to Siddhartha Sarkar.
Let $G_{m,n,k} = \mathbb{Z}_m \ltimes_k \mathbb{Z}_n$ be the split metacyclic group, where $k$ is a unit modulo $n$. We derive an upper bound for the diameter of $G_{m,n,k}$ using an arithmetic parameter called the \textit{weight}, which depends on $n$, $k$, and the order of $k$. As an application, we show how this would determine a bound on the diameter of an arbitrary metacyclic group.
In this paper, we are presenting a new approach to car theft detection and arresting system using RFID technology. The main purpose of this paper is to establish the concept and the architecture of the whole system. The system which we are able to develop still now is basically a demonstration model or a prototype of the system. The key thing in this system is the passive RFID tags which will be hidden inside the car and act as a unique identification number for the car. The information of all such tags will be maintained by a centralized server. In case of any car being theft, a report should be logged into the server by an authorized user and a network of RFID scanners installed in different check post, traffic signal or toll plazas in the city will search for the reported tag. Once it is found necessary security system would be activated for arresting the car.
Let $Γ(G,S)$ denote the Cayley graph of a group $G$ with respect to a set $S \subset G$. In this paper, we analyze the spectral properties of the Cayley graphs $\mathcal{T}_{m,n,k} = Γ(\mathbb{Z}_m \ltimes_k \mathbb{Z}_n, \{(\pm 1,0),(0,\pm 1)\})$, where $m,n \geq 3$ and $k^m \equiv 1 \pmod{n}$. We show that the adjacency matrix of $\mathcal{T}_{m,n,k}$, upto relabeling, is a block circulant matrix, and we also obtain an explicit description of these blocks. By extending a result due to Walker-Mieghem to Hermitian matrices, we show that $\mathcal{T}_{m,n,k}$ is not Ramanujan, when either $m > 8$, or $n \geq 400$.
The genus spectrum of a finite group $G$ is the set of all $g$ such that $G$ acts faithfully on a compact Riemann surface of genus $g$. It is an open problem to find a general description of the genus spectrum of the groups in interesting classes, such as the abelian $p$-groups. Motivated by the work of Talu for odd primes $p$, we develop a general combinatorial machinery, for arbitrary primes, to obtain a structured description of the so-called reduced genus spectrum of abelian $p$-groups. We have a particular view towards how to generally find the reduced minimum genus in this class of groups, determine the complete genus spectrum for a large subclass of abelian $p$-groups, consisting of those groups in a certain sense having `large' defining invariants, and use this to construct infinitely many counterexamples to Talu's Conjecture, saying that an abelian $p$-group is recoverable from its genus spectrum. Finally, we indicate the effectiveness of our combinatorial approach by applying it to quite a few explicit examples.
The genus spectrum of a finite group $G$ is the set of all $g\geq 2$ such that $G$ acts faithfully and orientation-preserving on a closed compact orientable surface of genus $g$. This article is an overview of some results relating the genus spectrum of $G$ to its group theoretical properties. In particular, the arithmetical properties of genus spectra are discussed, and explicit results are given on the 2-groups of maximal class, certain sporadic simple groups and a some of the groups PSL$(2,q)$, where $q$ is a small prime power. These results are partially new, and obtained through both theoretical reasoning and application of computational techniques.
A finite p-group is said to be of Gorenstein-Kulkarni type if the set of all elements of non-maximal order is a maximal subgroup. 2-groups of Gorenstein-Kulkarni type arise naturally in the study of group actions on compact Riemann surfaces. In this paper, we proceed towards a classification of p-groups of Gorenstein-Kulkarni type.