TAILIEUCHUNG - Characterizing rational groups whose irreducible characters vanish only on involutions

A rational group is a finite group whose irreducible complex characters are rational valued. The aim of this paper is to classify rational groups G for which every nonlinear irreducible character vanishes only on involutions. | Turk J Math (2015) 39: 408 – 411 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Characterizing rational groups whose irreducible characters vanish only on involutions Saeid JAFARI, Hesam SHARIFI∗ Department of Mathematics, Faculty of Science, Shahed University, Tehran, Iran Received: • Accepted/Published Online: • Printed: Abstract: A rational group is a finite group whose irreducible complex characters are rational valued. The aim of this paper is to classify rational groups G for which every nonlinear irreducible character vanishes only on involutions. Key words: Rational group, irreducible character, zero of character 1. Introduction Let G be a finite group and χ be a nonlinear irreducible ordinary character of G . A well-known theorem of Burnside states that there exists g ∈ G such that χ(g) = 0 ; such an element g is called a zero of χ, and we say χ vanishes on g . Zeroes of characters are important in finding the structure of Sylow subgroups of a finite group. Besides well-known theorems related to zeros of characters that usually appear in reference books, .[5], this subject has been well studied by many mathematicians such as Chillag [1]. An important result obtained by Moret´o and Navarro [9] applies when zeroes of characters occur on prime order elements. Dolfi et al. in [3] also proved that if p is a prime number and all of the p -elements of G are nonvanishing, then G has a normal Sylow p -subgroup. Throughout this paper, we use the following notations and terminologies. The order of the group G and the order of the element g ∈ G are denoted by

• Toán học
• Characterizing rational groups
• Rational group
• Irreducible characters vanish
• Irreducible character
• Zero of character