arXiv subjects

Rahul Dattatraya Kitture

Publications and source records attributed to Rahul Dattatraya Kitture.

5 recordsLinked to original sources

arXiv2026

Vanishing elements of prime power order and their class size property

Study of the structure of groups by variation in the arithmetic conditions on conjugacy classes and character degrees has produced several interesting results and open problems. In continuation of such work, Dolfi and Lucido in \cite{MR1826493} introduced a property for groups. For primes $p,q$, a group $G$ is said to have property $P(p,q)$ if every $p'$-element in $G$ has $q'$-class size. They obtained several results on the structure of $G$ and of some subgroups when $G$ satisfies the property $P(p,q)$. Motivated by this work, we introduce a vanishing analogue of the above property: for primes $p \neq q$, a finite group $G$ is said to have the property $P_v(p,q)$ if every vanishing $p'$-element of prime power order in $G$ has conjugacy class size not divisible by $q$. We show that no finite simple group satisfies the property $P_v(p,q)$ for primes $p\neq q$ dividing $|G|$. We use this result to show that if a finite group $G$ satisfies the property $P_v(p,q)$ with $p \neq q$ and $p > 2$, then $O^{q'}(G)$ (subgroup generated by all Sylow $q$-subgroups of $G$) is solvable. This generalises a result of Dolfi and Lucido under weaker conditions.

math.GR↗

arXiv2025

Vanishing Elements of Prime Power Order

An element $x$ in a finite group $G$ is said to be \textit{vanishing} if some (complex) irreducible character of $G$ takes value $0$ at $x$. In this article, we prove that every non-abelian finite simple group, except $\mathrm{SL}_2(4)$ and $\mathrm{SL}_2(8)$, contains a vanishing element \textit{of prime power order} whose conjugacy class size is divisible by three distinct primes. We use this result to obtain the following generalization of a result of Robati ($2021$): If $G$ is a non-solvable finite group in which, the conjugacy class size of all the vanishing elements of prime power order has at most two distinct prime divisors, then $G/\mathrm{Sol}(G)$ is a direct product of mutually isomorphic simple groups among $\mathrm{SL}_2(4)$ and $\mathrm{SL}_2(8)$. ($\mathrm{Sol}(G)$ is the largest normal solvable subgroup of $G$.)

math.GR↗

arXiv2017

Finite $p$-Groups of Nilpotency Class $3$ with Two Conjugacy Class Sizes

It is proved that, for a prime $p>2$ and integer $n\geq 1$, finite $p$-groups of nilpotency class $3$ and having only two conjugacy class sizes $1$ and $p^n$ exist if and only if $n$ is even; moreover, for a given even positive integer, such a group is unique up to isoclinism (in the sense of Philip Hall).

math.GR↗

arXiv2017

Finite Groups with Abelian Automorphism Groups: A Survey

In this article we present an extensive survey on the developments in the theory of non-abelian finite groups with abelian automorphism groups, and pose some problems and further research directions.

math.GR↗

arXiv2016

Note on Caranti's Method of Construction of Miller groups

The non-abelian groups with abelian group of automorphisms are widely studied. Following Earnley, such groups are called Miller groups, since the first example of such a group was given by Miller in 1913. Many other examples of Miller $p$-groups have been constructed by several authors. Recently, A. Caranti [{\it Israel J. Mathematics {\bf 205} (2015), 235-246}] provided module theoretic methods for constructing non-special Miller $p$-groups from special Miller $p$-groups. By constructing examples, we show that these methods do not always work. We also provide a sufficient condition on special Miller $p$-group for which the methods of Caranti work.

math.GR↗