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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 http://journals.tubitak.gov.tr/math/ doi:10.3906/mat-1408-4 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: 02.08.2014 • Accepted/Published Online: 14.01.2015 • Printed: 29.05.2015 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, e.g.[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 |G| and |g|, respectively. For the prime number p , Op (G) denotes the unique largest normal p -subgroup of G and E(pn ) denotes the elementary abelian p -group of order pn . For the elements x and g belonging the group G , by xg we mean g −1 xg . The symbol K : H stands for the semidirect .