arXiv subjects

Harish Kishnani

Publications and source records attributed to Harish Kishnani.

10 recordsLinked to original sources

arXiv2026

On Conjugacy Classes of Derangements in Symmetric and Alternating Groups

In this article, we prove two conjectures of Burness and Fusari [Timothy Burness and Marco Fusari, On derangements in simple permutation groups, Forum Math. Sigma 13 (2025)] concerning the powers and products of conjugacy classes of derangements in the symmetric and alternating groups: (1) We show that there exist two conjugacy classes $C$ and $D$ of derangements in $S_n$ such that $S_n=C^2\cup CD$, and (2) We show that there exists a conjugacy class $C$ of derangements in $A_n$ such that $C^2=A_n$, whenever $n\equiv 3\;(\text{mod}\;4)$. In fact, our result concerning the second conjecture holds in a considerably more general setting, which also answers affirmatively a question posed by Bertram [Edward Bertram, Even permutations as a product of two conjugate cycles, J. Comb. Theory, Ser. A 12 (1972), 368-380] in a particular case. Moreover, we show that any conjugacy class $C$ of derangements in $S_n$ (resp. $A_n$) contains a pair of elements that generate $S_n$ or $A_n$ (resp. $A_n$), unless $C$ is the conjugacy class of fixed-point-free involutions.

math.GR↗

arXiv2026

Waring Problem for matrices over finite local rings

This paper addresses the matrix Waring problem for matrices over finite principal local rings. Let $\mathcal{O}_{\ell}$ be a finite principal local ring of length $\ell$ with the maximal ideal $\mathfrak{m}$ and the residue field $\mathbb{F}_q = \mathcal{O}_\ell/\mathfrak{m}$. When $-1$ is a $k$-th power in $\mathbb{F}_q$ and the characteristic of $\mathbb{F}_q$ does not divide $k$, we show that for sufficiently large $q$, any matrix in $M_n(\mathcal{O}_\ell)$ can be expressed as a sum of two $k$-th powers. Furthermore, we establish that these two conditions are strictly necessary for the result to hold in general.

math.RA↗

arXiv2026

Images of Lie Polynomials on simple Lie algebras

A Lie polynomial is an element of a free Lie algebra $\mathcal F_k$ on $k$-generators, which defines a Lie map on a given Lie algebra $L$, by substituting $k$-elements of $L$. Similar to word maps on groups and polynomial maps on algebras, one studies here questions analogous to Waring-like problems, the L'vov-Kaplansky conjecture, etc. In this article, we would like to address a problem for Lie algebras parallel to the one Lubotzky solved (Images of word maps in finite simple groups, Glasg. Math. J., 56, no. 2, 465-469, 2014) for finite simple groups. It is easy to verify that the image of a Lie map is (a) closed under automorphism, and (b) contains $0$. In this article, we prove that for a simple Chevalley algebra over a finite field of ``very good'' characteristic, these two properties are enough to classify all possible subsets that can be the image of a Lie polynomial. The next question is to find such Lie polynomials for a given subset satisfying the two properties. Contrary to the results over an algebraically closed field, we find Lie polynomials in the case of Lie algebra $\mathfrak{sl}_2(q)$, for $q$ odd, which give each $\rm{GL}_2(q)$ conjugacy class together with zero as an image.

math.RA↗

arXiv2025

Defects in weighted graphs and Commutators

Let $R$ be a commutative ring. In \cite{KK_2025(1)}, the authors introduced $R$-weighted graphs as a tool for studying commutators in groups and Lie algebras. These graphs are equivalent to a system of balance equations, and their consistent labelings correspond to solutions of this system of balance equations. In this article, we apply these ideas in the case when $R$ is a field $F$. We focus on $F$-weighted graphs with four vertices and establish necessary and sufficient conditions for the existence of consistent labelings on them. A notion of defects in weighted graphs is introduced for this purpose. We prove that defects in weighted graphs prevent Lie brackets from being surjective onto its derived Lie subalgebra. Similarly, these defects prevent certain elements in the commutator subgroup of a nilpotent group of class $2$ from being a commutator. As an application of our techniques, we prove that for a Lie algebra $L$ whose dimension over $F$ is at most countable and the dimension of its derived subalgebra $L'$ is at most $3$, the Lie bracket is surjective onto $L'$. We provide a counterexample when $\dim(L') = 4$. We also characterize commutators among $L$' for the Lie algebras $L$ with $\dim(L/Z(L))\leq 4$.

math.CO↗

arXiv2025

$R$-weighted graphs and commutators

In this article, we introduce balance equations over commutative rings $R$ and associate $R$-weighted graphs to them so that solving balance equations corresponds to a consistent labeling of vertices of the associated graph. Our primary focus is the case when $R$ is a commutative local ring whose residue field contains at least three elements. In this case, we provide explicit solutions of balance equations. As an application, letting $R$ to be the ring of $p$-adic integers, we examine some necessary and sufficient conditions for a $p$-group of nilpotency class $2$ to have its set of commutators coincide with its commutator subgroup. We also apply our results to study the surjectivity of the Lie bracket in Lie algebras, without any restriction on their dimension and the underlined field.

math.CO↗

arXiv2023

Automorphic word maps and Amit--Ashurst conjecture

In this article, we address Amit--Ashurst conjecture on lower bounds of a probability distribution associated to a word on a finite nilpotent group. We obtain an extension of a result of Camina, Iniguez, and Thillaisundaram by providing improved bounds for the case of finite nilpotent groups of class $2$ for words in an arbitrary number of variables, and also settle the conjecture in some cases. We achieve this by obtaining words that are identically distributed on a group to a given word. In doing so, we also obtain an improvement of a result of Iniguez and Sangroniz using elementary techniques.

math.GR↗

arXiv2023

Products of conjugacy classes in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$

Let $k$ be a field with $u$-invariant $\leq2$. Assume further that $k$ is not quadratically closed, $\mathsf{char}(k)\neq 2$ and $|k|\geq 5$. It is known that the covering number of both $\text{SL}_2(k)$ and $\text{PSL}_2(k)$ is three, while their extended covering number is four. In this article, we completely describe the product of two conjugacy classes in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$. Further, we also describe the product of three conjugacy classes (at least two of which are distinct) in $\text{SL}_2(k)$ and $\text{PSL}_2(k)$.

math.GR↗

arXiv2022

Alternating groups as products of cycle classes - II

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev conjectured that $\lfloor \frac{2kl}{3} \rfloor \leq n(k,l)\leq \lfloor \frac{2kl}{3}\rfloor+1$. It is known that the conjecture holds when $k=2,3,4$. Moreover, it is also true when $3\mid l$. In this article, we determine the exact value of $n(k,l)$ when $3\nmid l$ and $k\geq 5$. As an immediate consequence, we get that $n(k,l)<\lfloor \frac{2kl}{3}\rfloor$ when $k\geq 5$, which shows that the above conjecture is not true in general. In fact, the difference between the exact value of $n(k,l)$ and the conjectured value grows linearly in terms of $k$. Our results also generalize the case of $k=2,3,4$.

math.CO↗

arXiv2022

Alternating groups as products of cycle classes

Given integers $k,l\geq 2$, where either $l$ is odd or $k$ is even, let $n(k,l)$ denote the largest integer $n$ such that each element of $A_n$ is a product of $k$ many $l$-cycles. In 2008, M. Herzog, G. Kaplan and A. Lev proved that if $k,l$ both are odd, $3\mid l$ and $l>3$, then $n(k,l)=\frac{2}{3}kl$. They further conjectured that if $k$ is even and $3\mid l$, then $n(k,l)=\frac{2}{3}kl+1$. In this article, we prove this conjecture. We also prove that $n(k,3)=2k+1$ if $k$ is odd.

math.CO↗

arXiv2022

Word Images and Their Impostors in Finite Nilpotent Groups

It was shown by Lubotzky in 2014 that automorphism invariant subsets of finite simple groups which contain identity are always word images. In this article, we study word maps on finite nilpotent groups and show that for arbitrary finite groups, the number of automorphism invariant subsets containing identity which are not word images, referred to as word image impostors, may be arbitrarily larger than the number of actual word images. In the course of it, we construct a $2$-exhaustive set of word maps on nilpotent groups of class $2$ and demonstrate its minimality in some cases.

math.GR↗