Isaacs-Navarro-Wolf conjecture for certain groups

Articles in Press

Document Type : Research Paper

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Department of Pure Mathematics, Faculty of Science, Imam Khomeini International University, Qazvin, Iran

Abstract

‎Let $G$ be a finite group‎ and let ${\rm F}_i(G)$ be the $i$-th term of the ascending Fitting series of $G$. In this paper‎, ‎we show that if $x$ is a non-vanishing element of non-nilpotent solvable group‎ $G$ and $G/{\rm F}_i(G)$ is supersolvable, then $x\in {\rm F}_i(G)$. Moreover, we prove that if $G$ is an extension of a supersolvable group by a nilpotent group and $x$ is a non-vanishing element, then $x$ lies in the Fitting subgroup $\mathrm{F}(G)$, which is an affirmative answer to the open conjecture stated in \cite{Isa2} by Isaacs, Navarro, and Wolf.

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References

[1] N. Ahanjideh and S. Mahmood Robati, The Isaacs-Navarro-Wolf conjecture for groups with one vanishing class size, Mediterr. J. Math., 20 No. 2 (2023) 101.
[2] D. Bubboloni, S. Dolfi and P. Spiga, Finite groups whose irreducible characters vanish only on p-elements, J. Pure Appl. Algebra, 213 No. 3 (2009) 370-376.
[3] S. Dolfi, E. Pacifici, L. Sanus and P. Spiga, On the vanishing prime graph of solvable groups, J. Group Theory, 13 (2010) 189-206.
[4] S. Dolfi, E. Pacifici and L. Sanus, Groups whose vanishing class sizes are not divisible by a given prime, Arch. Math., 94 No. 4 (2010) 311-317.
[5] K. Doerk and T. Hawkes, Finite Soluble Groups, Walter de Gruyter, Berlin/New York, 1992.
[6] I. M. Isaacs, G. Navarro and T. R. Wolf, Finite group elements where no irreducible character vanishes, J. Algebra, 222 No. 2 (1999) 413-423.
[7] S. Mahmood Robati, Groups whose set of vanishing elements is exactly a conjugacy class, Alg. Struc. Appl., 6 No. 2 (2019) 9-12.
[8] S. Mahmood Robati, Classification of groups whose vanishing elements are contained in exactly two conjugacy classes, Algebraic Struct. Appl., 11 No. 4 (2024) 305-310.
Algebraic Structures and Their Applications

Articles in Press, Corrected Proof
Available Online from 06 November 2026

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