arXiv subjects

Sonakshee Arora

Publications and source records attributed to Sonakshee Arora.

3 recordsLinked to original sources

arXiv2026

Forbidden subgraphs on conjugacy class graphs of groups

Let $G$ be a finite group. The commuting (resp. nilpotent) conjugacy class graph $Γ_{CCC}(G)$ (resp. $Γ_{NCC}(G)$) of $G$ is a simple graph whose vertex set consists of all non-central conjugacy classes of $G$, in which two distinct vertices $x^G$ and $y^G$ are adjacent if and only if there exist $a \in x^G$ and $b \in y^G$ such that $\langle a, b \rangle$ is an abelian (resp. nilpotent) subgroup. In this paper, we mainly investigate cographs, chordal graphs, split graphs, threshold graphs, and claw-free graphs in terms of forbidden induced subgraphs in $Γ_{CCC}(G)$ and $Γ_{NCC}(G)$. To be specific, we characterize the induced subgraphs in the commuting conjugacy class graph for symmetric groups, alternating groups, and sporadic groups. We also provide a complete classification of these properties for EPPO-groups, nilpotent groups, dihedral groups, dicyclic groups, and generalized dihedral groups in both commuting and nilpotent conjugacy class groups.

math.GR↗

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↗