Let G be a finite group with two transitive permutation representations on the sets Ω1 and Ω2, respectively. We are concerned with the case that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. We prove that if G has nilpotency class 2 then the permutation character π1 of G on Ω1 equals the permutation character π2 of G on Ω2. Furthermore, for these groups we prove that the stabilizer of a point in Ω1 is conjugate, under an automorphism of G, to the stabilizer of a point of Ω2. In Section 3 we present the following conjecture: Let G act primitively on Ω1 and on Ω2 and assume that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. Then the permutation character π1 of G on Ω1 and the permutation character π2 of G on Ω2 are comparable, i.e., if π1 ≠ π2 then either π1 - π2 or π2 - π1 is a character. We show that if the conjecture is false, then a minimal counterexample must be an almost simple group. Further results concerning other classes of groups are presented. © 2006 Elsevier Inc. All rights reserved

Spiga, P. (2006). Permutation characters and fixed-point-free elements in permutation groups. JOURNAL OF ALGEBRA, 299(1), 1-7 [10.1016/j.jalgebra.2006.03.015].

### Permutation characters and fixed-point-free elements in permutation groups

#### Abstract

Let G be a finite group with two transitive permutation representations on the sets Ω1 and Ω2, respectively. We are concerned with the case that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. We prove that if G has nilpotency class 2 then the permutation character π1 of G on Ω1 equals the permutation character π2 of G on Ω2. Furthermore, for these groups we prove that the stabilizer of a point in Ω1 is conjugate, under an automorphism of G, to the stabilizer of a point of Ω2. In Section 3 we present the following conjecture: Let G act primitively on Ω1 and on Ω2 and assume that the set of fixed-point-free elements of G on Ω1 coincides with the set of fixed-point-free elements of G on Ω2. Then the permutation character π1 of G on Ω1 and the permutation character π2 of G on Ω2 are comparable, i.e., if π1 ≠ π2 then either π1 - π2 or π2 - π1 is a character. We show that if the conjecture is false, then a minimal counterexample must be an almost simple group. Further results concerning other classes of groups are presented. © 2006 Elsevier Inc. All rights reserved
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Algebra and Number Theory
English
2006
299
1
1
7
none
Spiga, P. (2006). Permutation characters and fixed-point-free elements in permutation groups. JOURNAL OF ALGEBRA, 299(1), 1-7 [10.1016/j.jalgebra.2006.03.015].
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/10281/100584`
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