A Note on the Rational Non-linear Characters of Groups
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A Note on the Rational Non-linear Characters of Groups M. Norooz-Abadian1 · H. Sharifi1 Received: 1 February 2019 / Accepted: 1 August 2019 / © Springer Nature B.V. 2019
Abstract We study finite groups with rational valued non-linear characters and obtain the structure of these types of groups with derived subgroup of prime order. In particular, in the case of the size of derived subgroup of a group is two we conclude that this group is rational if and only if the group is a direct product of an extraspecial 2-group with an elementary abelian 2-group. Keywords Rational valued non-linear characters · Frobenius group Mathematics Subject Classification (2010) 20C15
1 Introduction Let us call a finite group G rational group or Q-group if every character χ ∈ Irr(G) is rational valued. We also call it is Q1 -group whenever every non-linear irreducible character is rational-valued. Elementary results on Q1 -groups are collected in the paper [3]. In particular, they characterized Q1 -groups by their vanishing-off subgroup, as introduced in [6]. In [8] authors investigated the structure of Frobenius Q1 -groups. Basmaji studied metabelian Q1 -groups [1]. Especially, he achieved some facts about Q1 -groups with cyclic derived subgroup. Motivated by [1, 7–9], we classify Q1 -groups with derived subgroup of prime order. Throughout the paper we consider finite groups, and we employ the following notation and terminology: The semi-direct product of group K with group H is denoted by K : H . A ∗ B is the central product of groups A and B, i.e., A ∗ B = AB with [A, B] = 1, where [A, B] is the commutator of A and B. The symbol Zn denotes a cyclic group of order n. For a prime p and a non-negative integer n, the symbol E(p n ) denotes the elementary abelian p-group of Presented by: Alistair Savage H. Sharifi
[email protected] 1
Department of Mathematics, Faculty of Science, Shahed University, Tehran, Iran
M. Norooz-Abadian, H. Sharifi
order p n , p a prime number; Q8 and D8 are employed to denote the quaternion and dihedral group of order 8, respectively. The main results of this work are the following Theorem 1.1 Suppose that G is a Q1 -group and its derived subgroup G is of order 2. Then G∼ = (A1 ∗A2 ∗. . .∗An )×E(2m ), where Ai , 1 ≤ i ≤ n, is isomorphic to one of the following (i) (ii) (iii) (iv) (v)
a, b | a 4 = b4 = 1, a b = a 3 ; a, b, c | a 4 = b4 = c2 = 1, [a, c] = [b, c] = 1, [a, b] = c; a, b, c | a 4 = b2 = c2 = 1, [a, c] = [b, c] = 1, [a, b] = c; Q8 ; D8 .
Theorem 1.2 Suppose that G is a Q1 -group such that |G | = p, where p is an odd prime. Then G is a Q1 -group if and only if one the following occurs (i) (ii)
G∼ = (Zp : Zp−1 ) × E(2n ) ∼ G = (Zp : Z2(p−1) ) × E(2n )
2 A Review and Preliminary Results We review some facts about rational groups. Let G be a finite group. Let nl(G) denote the set of non-linear irreducible characters of G. An element x ∈ G is called rational if χ (x) ∈ Q for every χ ∈ Irr(G). Also, χ ∈ Irr(G) is called a rational character if χ (x) ∈ Q for every x ∈ G. Lemma 2.1 ([5, p.
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