arXiv subjects
Publications and source records attributed to Ashish Kumar Das.
In this paper we study some of the basic properties of a graph which is constructed from the equivalence classes of non-zero zero-divisors determined by annihilator ideals of a poset. In particular, we demonstrate how this graph helps in identifying the annihilator prime ideals of a poset that satisfies the ascending chain condition for its proper annihilator ideals.
The commuting graph of a non-abelian group is a simple graph in which the vertices are the non-central elements of the group, and two distinct vertices are adjacent if and only if they commute. In this paper, we classify (up to isomorphism) all finite non-abelian groups whose commuting graphs are acyclic, planar or toroidal. We also derive explicit formulas for the genus of the commuting graphs of some well-known class of finite non-abelian groups, and show that, every collection of finite non-abelian groups whose commuting graphs have the same genus is finite.
The commutativity degree of a finite group is the probability that two randomly chosen group elements commute. The main object of this paper is to obtain a characterization for all finite groups of odd order with commutativity degree greater than or equal to 11/75.
This abstract presents (without proofs) some new results on commutativity degree of finite groups.
A finite group $G$ is said to be a POS-group if for each $ x $ in $G$ the cardinality of the set $\{y \in G | o(y) =o(x)\}$ is a divisor of the order of $G$. In this paper we study some of the properties of arbitrary POS-groups, and construct a couple of new families of nonabelian POS-groups. We also prove that the alternating group $A_n$, $n \ge 3$, is not a POS-group.
The object of this paper is to find a necessary and sufficient condition for the groups $G_1, G_2, ..., G_n$ so that every normal subgroup of the product $\prod_{i=1}^{n} G_i$ is of the type $\prod_{i=1}^{n} N_i$ with $N_i \trianglelefteq G_i$, $i=1,2, ..., n$. As a consequence we obtain a well-known result due to R. Remak about centreless completely reducible groups having finitely many direct factors.
It is well known that Dold and Milnor manifolds give generators for the unoriented bordism algebra ${\frak{N}}_*$ over ${\Bbb{Z}}_2$. The purpose of this paper is to determine those Milnor manifolds which represent the same bordism classes in ${\frak{N}}_*$ as their Dold counterparts.
This note proves that, for $F = \Bbb{R,C}$ or $\Bbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with $k < n$ and having real dimension $d$, constitute a linearly independent set in the unoriented bordism group ${\frak{N}}_d$ regarded as a ${\Bbb{Z}}_2$-vector space.