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Finite groups with gcd(χ(1), χc (1)) a prime

  • Li Gao , Zhongbi Wang and Guiyun Chen EMAIL logo
Published/Copyright: August 21, 2024
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Open Mathematics
From the journal Open Mathematics Volume 22 Issue 1

Abstract

The aim of this article is to study how the greatest common divisor of the degree and codegree of an irreducible character of a finite group influences its structure. We study a finite group G with gcd ( χ ( 1 ) , χ c ( 1 ) ) a prime for almost all irreducible characters χ of G , and obtain the following two conclusions:

  1. There does not exist any finite group G such that gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime, for each χ ∈ Irr ( G ) ♯ , where Irr ( G ) ♯ is the set of non-principal irreducible characters of G .

  2. Let G be a finite group, if gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime, for each χ ∈ Irr ( G ) \ Lin ( G ) , then G is solvable, where Lin ( G ) is the set of all linear irreducible characters of G .

Keywords: a finite group; an irreducible character; degree; codegree; the greatest common divisor; structure
MSC 2010: 20D10; 20D15; 20D20

1 Introduction

Let G be a finite group and χ ∈ Irr ( G ) , then the degree of χ is χ ( 1 ) . In 2007, Qian et al. [1] defined the codegree of χ as χ c ( 1 ) = ∣ G : ker χ ∣ χ ( 1 ) , denoted as χ c ( 1 ) , and did research on properties of codegrees of a finite group. Afterwards, there were systematic studies on codegrees and the relation between the codegree and structure of a group. Among them, similar to study of the set of degrees of all irreducible characters of G , a lot of studies on the set of codegrees of all irreducible characters of G , denoted as codeg ( G ) , were done. For example, Alizadeh et al. [2], Liu and Yang [3], Khosravi et al. [4] studied how ∣ codeg ( G ) ∣ influences the structure of G . Bahramian and Ahanjideh [5], Qian and Zhou [6], and Lv et al. [7] set up several equivalent characterizations of solvable groups depending on special relations among numbers in codeg ( G ) . Ahanjideh [8], Bahri et al. [9] gave a new characterization of some simple groups such as PSL ( 2 , q ) by codeg ( G ) . It is worth mentioning that Moretó [10] and Yang and Qian [11] studied the analog Huppert’s ρ − σ conjecture for character codegrees and have obtained beautiful results.

The study on the relationship between the greatest common divisor of degree and codegree of an irreducible character of a finite group started in [12], where Liang studied the finite groups G with coprime irreducible character degree and codegree. Inspired by [12], in this work, we study the finite groups with almost all greatest common divisors of its irreducible character degrees and codegrees being primes and prove the following two conclusions:

Theorem 1.1

There does not exist any finite group G such that gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime for each χ ∈ Irr ( G ) ♯ , where Irr ( G ) ♯ is the set of non-principal irreducible characters of G.

Theorem 1.2

Let G be a finite group, if gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime for each χ ∈ Irr ( G ) \ Lin ( G ) , then G is solvable, where Lin ( G ) is the set of all linear irreducible characters of G.

2 Preliminaries

Lemma 2.1

[13, Theorem 3] If n ≥ 6 , then Irr ( A n ) contains characters of degree n ( n − 3 ) ⁄ 2 and ( n − 1 ) ( n − 2 ) ⁄ 2 that extend to S n .

Lemma 2.2

[14, Theorem A] Let S = PSL ( 2 , q ) , where q = p f > 3 for a prime p , A = Aut ( S ) , and let S ≤ H ≤ A . Set G = PGL ( 2 , q ) if δ ¯ ∈ H and G = S if δ ¯ ∉ H , and let ∣ H : G ∣ = d = 2 a m , m is odd. If p is odd, let ε = ( − 1 ) ( q − 1 ) ⁄ 2 . The set of irreducible character degrees of H is

c d ( H ) = { 1 , q , ( q + ε ) ⁄ 2 } ∪ { ( q − 1 ) 2 a i : i ∣ m } ∪ { ( q + 1 ) j : j ∣ d }

with the following exceptions:

  1. If p is odd with H ≰ S ⟨ φ ⟩ or if p = 2 , then ( q + ε ) ⁄ 2 is not a degree of H;

  2. If f is odd, p = 3 , and H = S ⟨ φ ⟩ , then i ≠ 1 ;

  3. If f is odd, p = 3 , and H = A , then j ≠ 1 ;

  4. If f is odd, p = 2 , 3, or 5, and H = S ⟨ φ ⟩ , then j ≠ 1 ;

  5. If f ≡ 2 (mod 4), p = 2 , 3, and H = S ⟨ φ ⟩ or H = gcd ⟨ δ ¯ φ ⟩ , then j ≠ 2 ;

where φ is a field automorphism of PSL ( 2 , q ) with order f , δ ¯ is an automorphism of PSL ( 2 , q ) induced by a diagonal automorphism.

Lemma 2.3

[1, Lemma 2.1] Let G be a finite group, if N is a subnormal subgroup of G, ψ ∈ Irr ( N ) , then ψ c ( 1 ) ∣ χ c ( 1 ) , for all χ ∈ Irr ( G ∣ ψ ) .

In order to prove the theorems, we need the degrees of irreducible characters of non-abelian simple groups, which can be found, e.g., in [7,15–21]. For convenience, we use the information from these references to create Tables 1, 2, 3, in which we list a special irreducible character χ ( 1 ) such that gcd ( χ ( 1 ) , χ c ( 1 ) ) is not a prime.

Table 1

Irreducible character degree of a sporadic group ≠ J 1

Group χ ( 1 ) Factor of gcd ( χ ( 1 ) , χ c ( 1 ) ) Group χ ( 1 ) Factor of gcd ( χ ( 1 ) , χ c ( 1 ) )
M 11 2 2 ⋅ 11 2 2 M 12 2 4 2 2
M 22 2 ⋅ 3 ⋅ 5 ⋅ 7 2 ⋅ 3 J 2 2 2 ⋅ 3 2 2 2 ⋅ 3
M 23 2 3 ⋅ 11 ⋅ 23 2 3 H S 2 ⋅ 5 ⋅ 7 ⋅ 11 2 ⋅ 5
J 3 2 2 ⋅ 3 4 2 2 ⋅ 3 M 24 2 2 ⋅ 3 2 ⋅ 7 2 2 ⋅ 3
M c L 2 ⋅ 5 ⋅ 7 ⋅ 11 2 ⋅ 5 H e 2 3 ⋅ 5 ⋅ 17 2 3 ⋅ 5
R u 2 2 ⋅ 3 2 ⋅ 7 ⋅ 13 2 2 ⋅ 3 S u z 2 2 ⋅ 7 ⋅ 13 2 2
O ′ N 2 6 ⋅ 3 2 ⋅ 19 2 3 ⋅ 3 2 C o 3 2 7 ⋅ 7 2 3
C o 2 2 3 ⋅ 11 ⋅ 23 2 3 F i 22 2 ⋅ 3 ⋅ 13 2 ⋅ 3
H N 2 3 ⋅ 5 ⋅ 19 2 3 ⋅ 5 L y 2 4 ⋅ 5 ⋅ 31 2 4 ⋅ 5
T h 2 3 ⋅ 31 2 3 F i 23 2 2 ⋅ 3 ⋅ 13 ⋅ 23 2 2 ⋅ 3
C o 1 2 2 ⋅ 3 ⋅ 23 2 2 ⋅ 3 J 4 2 ⋅ 3 2 ⋅ 31 ⋅ 37 ⋅ 43 2 ⋅ 3
F i 24 ′ 3 ⋅ 7 2 ⋅ 17 ⋅ 23 3 ⋅ 7 B 3 2 ⋅ 5 ⋅ 23 ⋅ 31 3 2 ⋅ 5
M 2 2 ⋅ 31 ⋅ 41 ⋅ 59 ⋅ 71 2 2
Table 2

Irreducible character degree of simple groups of Lie type ≠ A 1 ( q )

Group χ ( 1 ) Factor of g c d ( χ ( 1 ) , χ c ( 1 ) )
A 2 ( q ) ( q > 2 ) q ( q + 1 ) 2 q ( q odd)
q ( q even)
A 3 ( q ) ( q > 2 ) q 3 ( q 2 + q + 1 ) q 3
A 4 ( q ) q ( q + 1 ) ( q 2 + 1 ) q ( q + 1 )
A n ( q ) ( n > 4 ) 1 ( q ‒ 1 ) ( q 2 ‒ 1 ) q 3 ( q n ‒ 1 ‒ 1 ) ( q n ‒ 1 ) q 3
B 2 ( q ) ( q = p r , p is odd) 1 2 q ( q ‒ 1 ) 2 2 q
B 2 ( q ) ( q > 2 ) ( q = 2 r ) 1 2 q ( q ‒ 1 ) 2 q
B n ( q ) ( n > 2 ) 1 2 ( q 2 ‒ 1 ) 2 q 4 ( q n ‒ 2 ‒ 1 ) ( q n ‒ 1 ‒ 1 ) ( q n ‒ 1 + 1 ) ( q n + 1 ) q 2
C n ( q ) ( n ≥ 3 ) 1 ( q 2 ‒ 1 ) 2 q 3 ( q 2 ( n ‒ 2 ) ‒ 1 ) ( q 2 n ‒ 1 ) q 2
D 4 ( 2 ) 28 2 2
D 4 ( 3 ) 260 2 2
D 4 ( q ) ( q > 3 ) 1 2 q 3 ( q + 1 ) 4 ( q 2 ‒ q + 1 ) q 2
D 5 ( q ) ( q ‒ 1 ) 2 ( q 4 ‒ 1 ) ( q 5 ‒ 1 ) ( q 6 ‒ 1 ) ( q 2 ‒ 1 ) ( q 2 + 1 )
D n ( q ) ( n > 5 ) 1 ( q 2 ‒ 1 ) 2 ( q 4 ‒ 1 ) q 6 ( q n ‒ 4 + 1 ) ( q 2 ( n ‒ 3 ) ‒ 1 ) ( q 2 ( n ‒ 1 ) ‒ 1 ) ( q n ‒ 1 ) q 2
A 2 2 ( q ) ( q = 2 r , r > 2 ) q ( q ‒ 1 ) 2 r
A 2 2 ( q ) ( q = p r , p is odd) q ( q ‒ 1 ) 2 q
A 3 2 ( q ) q 2 ( q 2 + 1 ) q 2
A n 2 ( q ) ( n > 3 ) 1 ( q + 1 ) ( q 2 ‒ 1 ) q 3 ( q n ‒ 1 ‒ ( ‒ 1 ) n ‒ 1 ) ( q n ‒ ( ‒ 1 ) n ) q 3
D 4 2 ( q ) 1 2 q 3 ( q + 1 ) 4 ( q 2 ‒ q + 1 ) q 3
D n 2 ( q ) ( n > 4 ) 1 2 ( q 2 + 1 ) ( q ‒ 1 ) 2 q 3 ( q n ‒ 3 ‒ 1 ) ( q n ‒ 2 + 1 ) ( q n ‒ 1 ‒ 1 ) ( q n + 1 ) q 3
Table 3

Irreducible character degree of simple groups of Lie type ≠ A 1 ( q ) – continued

Group χ ( 1 ) Factor of g c d ( χ ( 1 ) , χ c ( 1 ) )
G 2 ( 3 ) 78 6
G 2 ( 4 ) 78 6
G 2 ( q ) ( q = p r , p is odd) 1 6 q ( q ‒ 1 ) 2 ( q 2 ‒ q + 1 ) 2 q
G 2 ( q ) ( q = 2 r , r > 2 ) 1 6 q ( q ‒ 1 ) 2 ( q 2 ‒ q + 1 ) q
F 4 ( q ) q 2 ( q 2 + q + 1 ) 2 ( q 2 ‒ q + 1 ) 2 ( q 4 ‒ q 2 + 1 ) q 2
E 6 ( q ) q 4 ϕ 2 3 ϕ 4 2 ϕ 6 2 ϕ 8 ϕ 12 q 4
E 7 ( q ) 1 2 q 3 ϕ 1 4 ϕ 3 2 ϕ 5 ϕ 7 ϕ 9 ϕ 14 q 2
E 8 ( q ) 1 2 q 4 ϕ 1 4 ϕ 3 2 ϕ 4 2 ϕ 5 2 ϕ 7 ϕ 9 ϕ 10 ϕ 12 ϕ 15 ϕ 20 ϕ 30 q 3
D 4 3 ( q 3 ) ϕ 1 ϕ 2 ϕ 3 ϕ 6 2 ϕ 12 ϕ 1 ϕ 2 ϕ 3
E 6 2 ( q 2 ) q 6 ϕ 3 2 ϕ 6 3 ϕ 12 ϕ 18 q 6
F 4 2 ( 2 ) ′ 78 6
F 4 2 ( q 2 ) ( q 2 > 2 ) q 2 ϕ 12 ϕ 24 q 2
B 2 2 ( q ) ( q = 2 2 m + 1 , m > 1 ) ( q ‒ 1 ) ( q 2 ) 1 2 2 m
G 2 2 ( 27 ) 3 3 ⋅ 19 ⋅ 37 3 3
G 2 2 ( q ) ( q = 3 2 m + 1 , m > 1 ) ( q 3 ) 1 2 ϕ 1 ϕ 2 ϕ 4 3 m

where ϕ n is the value of n ‒ t h cyclotomic polynomial at q .

3 Proofs

Proof of Theorem 1.1

Since gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime for χ ∈ Irr ( G ) ♯ , it follows that χ ( 1 ) > 1 for χ ∈ Irr ( G ) ♯ and ∣ G : G ′ ∣ = 1 . Therefore, G = G ′ and G is a nonsolvable group, there exists N ⊴ G , such that G ⁄ N is a non-abelian simple group. Let S = G ⁄ N , then gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime for χ ∈ Irr ( S ) ♯ . We prove that this is impossible for any non-abelian simple group S , and thus, Theorem 1.1 follows.

If S is a sporadic group, using Table 1, we can see that for a sporadic group ≠ J 1 , there always exists an irreducible character χ such that gcd ( χ ( 1 ) , χ c ( 1 ) ) is a composite number, which is a contradiction. Hence S ≅ J 1 , but Irr ( J 1 ) has a character χ of degree 7 ⋅ 11 , which implies gcd ( χ ( 1 ) , χ c ( 1 ) ) = 1 , which is a contradiction.

If S is a simple group of Lie type ≠ F 4 2 ( 2 ) ′ , let χ be the Steinberg character, then χ ( 1 ) is coprime to χ c ( 1 ) , which is a contradiction. If S ≅ F 4 2 ( 2 ) ′ ( 2 11 ⋅ 3 3 ⋅ 5 2 ⋅ 13 ) , then S has a character χ of degree 27 , thus gcd ( χ ( 1 ) , χ c ( 1 ) ) = 1 , which is a contradiction.

Finally, assume that S is an alternating group A n . While n = 5 , 6 , or 7, checking the character tables of A 5 , A 6 , and A 7 , we can see that S does not satisfy the condition of Theorem 1.1. While n ≥ 8 , by Lemma 2.1, S has an irreducible character χ of degree n ( n − 3 ) ⁄ 2 and another one, say, φ of degree ( n − 1 ) ( n − 2 ) ⁄ 2 , so χ c ( 1 ) = ( n − 1 ) ( n − 2 ) [ ( n − 4 ) ! ] , φ c ( 1 ) = n [ ( n − 3 ) ! ] .

If n = 4 m , then m ≥ 2 , and χ ( 1 ) = 2 m ( 4 m − 3 ) , χ c ( 1 ) = ( 4 m − 1 ) ( 4 m − 2 ) [ ( 4 m − 4 ) ! ] . Since 2 m ≤ 4 ( m − 4 ) , it follows that 2 m ∣ gcd ( χ ( 1 ) , χ c ( 1 ) ) , which is a contradiction.

If n = 4 m + 1 , then m ≥ 2 , and φ ( 1 ) = 2 m ( 4 m − 1 ) , φ c ( 1 ) = ( 4 m + 1 ) [ ( 4 m − 2 ) ! ] . Since 2 m < ( 4 m − 2 ) , it follows that 2 m ∣ gcd ( φ ( 1 ) , φ c ( 1 ) ) , which is a contradiction.

If n = 4 m + 2 , then m ≥ 2 , and φ ( 1 ) = 2 m ( 4 m + 1 ) , φ c ( 1 ) = ( 4 m + 2 ) [ ( 4 m − 1 ) ! ] . Since 2 m < ( 4 m − 1 ) , it follows that 2 m ∣ gcd ( φ ( 1 ) , φ c ( 1 ) ) , which is a contradiction.

If n = 4 m + 3 , then m ≥ 2 , and χ ( 1 ) = 2 m ( 4 m + 3 ) , χ c ( 1 ) = ( 4 m + 2 ) ( 4 m + 1 ) [ ( 4 m − 1 ) ! ] . Since 2 m < ( 4 m − 1 ) , it follows that 2 m ∣ gcd ( χ ( 1 ) , χ c ( 1 ) ) , which is a contradiction. Theorem 1.1 is proved.□

Proof of Theorem 1.2

Let G be a minimal counter example. According to Theorem 1.1, G is not a simple group. Assume that N is a minimal normal subgroup of G . Since the hypothesis is inherited by quotients, N is a non-solvable group. Assume N = S 1 × S 2 × … × S r , where S i are isomorphic non-abelian simple groups, say S i ≅ S .

Since the class of solvable groups is a saturated formation, it follows from induction that N is the unique minimal normal subgroup of G . Note that S is a subnormal subgroup of G , according to Lemma 2.3, for any θ ∈ Irr ( S ) , gcd ( θ ( 1 ) , θ c ( 1 ) ) must be equal to 1 or a prime.

If S is a sporadic simple group, then it follows by Table 1 and Theorem 1.1 that S ≅ J 1 ( ∣ J 1 ∣ = 2 3 ⋅ 3 ⋅ 5 ⋅ 7 ⋅ 11 ⋅ 19 ) , we assert r = 1 . Otherwise N ≥ J 1 × J 1 , take χ 1 , χ 2 ∈ Irr ( J 1 ) with χ 1 ( 1 ) = 2 3 ⋅ 7 , χ 2 ( 1 ) = 7 ⋅ 11 , then χ = χ 1 × χ 2 ∈ Irr ( N ) , χ ( 1 ) = χ 1 ( 1 ) χ 2 ( 1 ) = 2 3 ⋅ 7 2 ⋅ 11 , and gcd ( χ ( 1 ) , χ c ( 1 ) ) = 2 3 ⋅ 11 , which is a contradiction. Till now, we have proved that N ≅ J 1 is the only minimal normal subgroup of G , so C G ( J 1 ) = 1 . Hence, N ⁄ C Theorem yields G ≤ Aut ( J 1 ) = J 1 . Therefore, G ≅ J 1 , which is a contradiction.

If S is an alternating group A n , n ≥ 5 , then by the proof of Theorem 1.1, S cannot be A n ( n ≥ 8 ) . If S ≅ A 7 , then S has a character θ with θ ( 1 ) = 6 , and gcd ( θ ( 1 ) , θ c ( 1 ) ) = 6 , which is a contradiction. If S ≅ A 5 , we assert N ≅ A 5 . Otherwise N ≥ A 5 × A 5 , let θ 1 , θ 2 ∈ Irr ( A 5 ) , and θ 1 ( 1 ) = 3 , θ 2 ( 1 ) = 5 , then θ = θ 1 × θ 2 ∈ Irr ( N ) , θ ( 1 ) = θ 1 ( 1 ) θ 2 ( 1 ) = 15 , and gcd ( θ ( 1 ) , θ c ( 1 ) ) = 15 , which is a contradiction. Therefore, N ≅ A 5 is the unique minimal normal subgroup of G , hence C G ( A 5 ) = 1 and G ≤ Aut ( A 5 ) = S 5 , one has A 5 ⊴ G ≤ S 5 , thus G = A 5 , or S 5 , a contradiction. If S ≅ A 6 , then by the same reasoning N ≅ A 6 , and G ≅ S 6 , PSO ( 3 , 9 ) ( 2 4 ⋅ 3 2 ⋅ 5 ) , M 10 ( 2 4 ⋅ 3 2 ⋅ 5 ) or PΓO ( 3 , 9 ) ( 2 5 ⋅ 3 2 ⋅ 5 ) . Obviously G cannot be S 6 . If G ≅ PSO ( 3 , 9 ) , M 10 , or PΓO ( 3 , 9 ) , then G has an irreducible character χ of degree 9 so that gcd ( χ ( 1 ) , χ c ( 1 ) ) = 1 , which is a contradiction.

If S is a simple group of Lie type, it follows by Tables 2 and 3 that S ≅ PSL ( 2 , q ) , q = p r , where p is a prime. Similar reasoning as previous paragraphs, we can prove that N ≅ PSL ( 2 , q ) and N ≤ G ≤ Aut ( PSL ( 2 , q ) ) = PΓL ( 2 , q ) . Thus,

G ⁄ N ≤ Aut ( PSL ( 2 , q ) ) ⁄ N = C r , p = 2 , C r × C 2 , p ≠ 2 .

We assert that any nonlinear irreducible character of G is faithful. Otherwise there exists some χ ∈ Irr ( G ) with ker χ ≠ 1 , then it follows by uniqueness of N that N ≤ ker χ and χ ( 1 ) = 1 , which is a contradiction.

We have two steps more to finish the proof.

Assume p = 2 . Then, N = PSL ( 2 , q ) = PGL ( 2 , q ) , q = 2 r . By Lemma 2.2 and notations there, we have d = ∣ G : PGL ( 2 , q ) ∣ = 2 a m , where m is odd, and there exists χ ∈ Irr ( G ) with χ ( 1 ) = q , then χ c ( 1 ) = ∣ G ∣ ⁄ χ ( 1 ) = 2 a m ( q + 1 ) ( q − 1 ) . As gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime, gcd ( χ ( 1 ) , χ c ( 1 ) ) = gcd ( q , 2 a m ( q + 1 ) ( q − 1 ) ) = gcd ( q , 2 a m ) = 2 , that is, a = 1 . If m > 1 , then ( q − 1 ) 2 i ∈ c d ( G ) , where i runs over factors of m . Now we take i = 1 , there is ϕ ( 1 ) = 2 ( q − 1 ) ∈ c d ( G ) , then ϕ c ( 1 ) = ∣ G ∣ ⁄ ϕ ( 1 ) = m q ( q + 1 ) . As gcd ( ϕ ( 1 ) , ϕ c ( 1 ) ) is a prime, one has gcd ( ϕ ( 1 ) , ϕ c ( 1 ) ) = gcd ( 2 ( q − 1 ) , m q ( q + 1 ) ) = gcd ( 2 ( q − 1 ) , m q ) = 2 , thus gcd ( q − 1 , m ) = 1 . Take i = m > 1 , there is η ( 1 ) = 2 m ( q − 1 ) ∈ c d ( G ) , and η c ( 1 ) = ∣ G ∣ ⁄ η ( 1 ) = q ( q + 1 ) . As gcd ( η ( 1 ) , η c ( 1 ) ) is a prime, we obtain gcd ( η ( 1 ) , η c ( 1 ) ) = g c d ( 2 m ( q − 1 ) , q ( q + 1 ) ) = g c d ( 2 m , q ( q + 1 ) ) = 2 , so gcd ( q + 1 , m ) = 1 . For 1 < i < m , there is φ ( 1 ) = ( q − 1 ) 2 i ∈ c d ( G ) , so φ c ( 1 ) = ∣ G ∣ ⁄ φ ( 1 ) = m q ( q + 1 ) ⁄ i . Again by gcd ( φ ( 1 ) , φ c ( 1 ) ) is a prime we come to gcd ( φ ( 1 ) , φ c ( 1 ) ) = gcd ( 2 i ( q − 1 ) , m q ( q + 1 ) ⁄ i ) = gcd ( 2 i , m q ⁄ i ) = 2 , which forces gcd ( i , m ⁄ i ) = 1 . Again by Lemma 2.2, there is ξ ( 1 ) = ( q + 1 ) j ∈ c d ( G ) , where j runs over factors of m . Take j > 1 , one has ξ c ( 1 ) = d q ( q − 1 ) ⁄ j = 2 m q ( q − 1 ) ⁄ j , and gcd ( ξ ( 1 ) , ξ c ( 1 ) ) = gcd ( ( q + 1 ) j , 2 m q ( q − 1 ) ⁄ j ) = gcd ( j , 2 m q ( q − 1 ) ⁄ j ) = gcd ( j , ( q − 1 ) ) = 1 , which is a contradiction. Therefore, m = 1 , ∣ G ∣ = 2 q ( q + 1 ) ( q − 1 ) . Again by Lemma 2.2, c d ( G ) = { 1 , q , 2 ( q − 1 ) , q + 1 } or { 1 , q , 2 ( q − 1 ) , q + 1 , 2 ( q + 1 ) } . Take χ ( 1 ) = q + 1 , then gcd ( χ ( 1 ) , χ c ( 1 ) ) = 1 , which is a contradiction.

Assume p is odd. By Lemma 2.2 and notations there, if δ ¯ ∈ G , where δ ¯ is the automorphism of PSL ( 2 , q ) induced by diagonal automorphism, then d = ∣ G : PGL ( 2 , q ) ∣ = 2 a m , where m is odd. At this moment, we consider χ ( 1 ) = q ∈ c d ( G ) and obtain χ c ( 1 ) = ∣ G ∣ ⁄ χ ( 1 ) = 2 a m ( q + 1 ) ( q − 1 ) , then gcd ( χ ( 1 ) , χ c ( 1 ) ) = gcd ( q , 2 a m ( q + 1 ) ( q − 1 ) ) = gcd ( q , m ) = p for gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime, so p ‖ m , further m > 1 . Again by Lemma 2.2, G has irreducible characters φ such that φ ( 1 ) = ( q − 1 ) 2 a i , where i ∣ m . Set i = m and η ( 1 ) = 2 a m ( q − 1 ) ∈ c d ( G ) , then gcd ( η ( 1 ) , η c ( 1 ) ) = gcd ( 2 a m ( q − 1 ) , q ( q + 1 ) ) . Obviously, 2 p ∣ gcd ( η ( 1 ) , η c ( 1 ) ) , which is a contradiction.

If δ ¯ ∉ G , then d = ∣ G : PSL ( 2 , q ) ∣ = 2 a m , where m is an odd number. Let χ ∈ Irr ( G ) with χ ( 1 ) = q , then χ c ( 1 ) = ∣ G ∣ ⁄ χ ( 1 ) = 2 a m ( q + 1 ) ( q − 1 ) ⁄ 2 . gcd ( χ ( 1 ) , χ c ( 1 ) ) = gcd ( q , 2 a m ( q + 1 ) ( q − 1 ) ⁄ 2 ) = gcd ( q , m ) = p as gcd ( χ ( 1 ) , χ c ( 1 ) ) is a prime, so p ‖ m , further, m > 1 . By Lemma 2.2, G has irreducible characters φ with φ ( 1 ) = ( q − 1 ) 2 a i , where i ∣ m . Set i = m and η ( 1 ) = 2 a m ( q − 1 ) ∈ c d ( G ) , then η c ( 1 ) = ∣ G ∣ ⁄ η ( 1 ) = q ( q + 1 ) ⁄ 2 , and gcd ( η ( 1 ) , η c ( 1 ) ) = gcd ( 2 a m ( q − 1 ) , q ( q + 1 ) ⁄ 2 ) . If 4 ∣ ( q + 1 ) , then 2 p ∣ gcd ( η ( 1 ) , η c ( 1 ) ) , which is a contradiction. Thus 2 ‖ ( q + 1 ) , which forces 4 ∣ ( q − 1 ) . Now by Lemma 2.2 again, G has an irreducible character ψ with ψ ( 1 ) = m ( q + 1 ) , so ψ c ( 1 ) = ∣ G ∣ ⁄ ψ ( 1 ) = 2 a q ( q − 1 ) ⁄ 2 , and gcd ( ψ ( 1 ) , ψ c ( 1 ) ) = gcd ( m ( q + 1 ) , 2 a q ( q − 1 ) ⁄ 2 ) , which implies 2 p ∣ gcd ( ψ ( 1 ) , ψ c ( 1 ) ) , which is a contradiction.

Till now, we have proved that S cannot be any non-abelian simple group, so N is an elementary abelian group. Since the hypothesis is inherited by quotients and G is a minimal counterexample, G ⁄ N is a solvable group, so is G . This concludes Theorem 1.2.□

Acknowledgements

The authors are grateful for the reviewer’s valuable comments that improved the manuscript.

  1. Funding information: This work was supported by National Natural Science Foundation of China (Grant No. 12071376) and Chongqing Technology and Business University (2353005).

  2. Author contributions: All authors have accepted responsibility for the entire content of this manuscript and consented to its submission to the journal, reviewed all the results, and approved the final version of the manuscript.

  3. Conflict of interest: The authors state no conflict of interest.

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Received: 2023-11-01
Revised: 2024-03-19
Accepted: 2024-07-17
Published Online: 2024-08-21

© 2024 the author(s), published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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  2. On the maximum atom-bond sum-connectivity index of graphs
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  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
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https://doi.org/10.1515/math-2024-0037
Keywords for this article
a finite group; an irreducible character; degree; codegree; the greatest common divisor; structure
Creative Commons
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Articles in the same Issue

  1. Special Issue on Contemporary Developments in Graph Topological Indices
  2. On the maximum atom-bond sum-connectivity index of graphs
  3. Upper bounds for the global cyclicity index
  4. Zagreb connection indices on polyomino chains and random polyomino chains
  5. On the multiplicative sum Zagreb index of molecular graphs
  6. The minimum matching energy of unicyclic graphs with fixed number of vertices of degree two
  7. Special Issue on Convex Analysis and Applications - Part I
  8. Weighted Hermite-Hadamard-type inequalities without any symmetry condition on the weight function
  9. Scattering threshold for the focusing energy-critical generalized Hartree equation
  10. (p, q)-Compactness in spaces of holomorphic mappings
  11. Characterizations of minimal elements of upper support with applications in minimizing DC functions
  12. Some new Hermite-Hadamard-type inequalities for strongly h-convex functions on co-ordinates
  13. Global existence and extinction for a fast diffusion p-Laplace equation with logarithmic nonlinearity and special medium void
  14. Extension of Fejér's inequality to the class of sub-biharmonic functions
  15. On sup- and inf-attaining functionals
  16. Regularization method and a posteriori error estimates for the two membranes problem
  17. Rapid Communication
  18. Note on quasivarieties generated by finite pointed abelian groups
  19. Review Articles
  20. Amitsur's theorem, semicentral idempotents, and additively idempotent semirings
  21. A comprehensive review of the recent numerical methods for solving FPDEs
  22. On an Oberbeck-Boussinesq model relating to the motion of a viscous fluid subject to heating
  23. Pullback and uniform exponential attractors for non-autonomous Oregonator systems
  24. Regular Articles
  25. On certain functional equation related to derivations
  26. The product of a quartic and a sextic number cannot be octic
  27. Combined system of additive functional equations in Banach algebras
  28. Enhanced Young-type inequalities utilizing Kantorovich approach for semidefinite matrices
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  50. Complex dynamics of a nonlinear discrete predator-prey system with Allee effect
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  54. Existence and properties of soliton solution for the quasilinear Schrödinger system
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  57. Matrix stretching
  58. A singular perturbation result for a class of periodic-parabolic BVPs
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  62. Variational approach to Kirchhoff-type second-order impulsive differential systems
  63. Convergence rate of the truncated Euler-Maruyama method for highly nonlinear neutral stochastic differential equations with time-dependent delay
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  65. The limit theorems on extreme order statistics and partial sums of i.i.d. random variables
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  67. Solving multi-point problem for Volterra-Fredholm integro-differential equations using Dzhumabaev parameterization method
  68. Finite groups with gcd(χ(1), χc (1)) a prime
  69. Small values and functional laws of the iterated logarithm for operator fractional Brownian motion
  70. The hull-kernel topology on prime ideals in ordered semigroups
  71. ℐ-sn-metrizable spaces and the images of semi-metric spaces
  72. Strong laws for weighted sums of widely orthant dependent random variables and applications
  73. An extension of Schweitzer's inequality to Riemann-Liouville fractional integral
  74. Construction of a class of half-discrete Hilbert-type inequalities in the whole plane
  75. Analysis of two-grid method for second-order hyperbolic equation by expanded mixed finite element methods
  76. Note on stability estimation of stochastic difference equations
  77. Trigonometric integrals evaluated in terms of Riemann zeta and Dirichlet beta functions
  78. Purity and hybridness of two tensors on a real hypersurface in complex projective space
  79. Classification of positive solutions for a weighted integral system on the half-space
  80. A quasi-reversibility method for solving nonhomogeneous sideways heat equation
  81. Higher-order nonlocal multipoint q-integral boundary value problems for fractional q-difference equations with dual hybrid terms
  82. Noetherian rings of composite generalized power series
  83. On generalized shifts of the Mellin transform of the Riemann zeta-function
  84. Further results on enumeration of perfect matchings of Cartesian product graphs
  85. A new extended Mulholland's inequality involving one partial sum
  86. Power vector inequalities for operator pairs in Hilbert spaces and their applications
  87. On the common zeros of quasi-modular forms for Γ+0(N) of level N = 1, 2, 3
  88. One special kind of Kloosterman sum and its fourth-power mean
  89. The stability of high ring homomorphisms and derivations on fuzzy Banach algebras
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  91. Commutators of multilinear fractional maximal operators with Lipschitz functions on Morrey spaces
  92. Vector optimization problems with weakened convex and weakened affine constraints in linear topological spaces
  93. The curvature entropy inequalities of convex bodies
  94. Brouwer's conjecture for the sum of the k largest Laplacian eigenvalues of some graphs
  95. High-order finite-difference ghost-point methods for elliptic problems in domains with curved boundaries
  96. Riemannian invariants for warped product submanifolds in Q ε m × R and their applications
  97. Generalized quadratic Gauss sums and their 2mth power mean
  98. Euler-α equations in a three-dimensional bounded domain with Dirichlet boundary conditions
  99. Enochs conjecture for cotorsion pairs over recollements of exact categories
  100. Zeros distribution and interlacing property for certain polynomial sequences
  101. Random attractors of Kirchhoff-type reaction–diffusion equations without uniqueness driven by nonlinear colored noise
  102. Study on solutions of the systems of complex product-type PDEs with more general forms in ℂ2
  103. Dynamics in a predator-prey model with predation-driven Allee effect and memory effect
  104. A note on orthogonal decomposition of 𝔰𝔩n over commutative rings
  105. On the δ-chromatic numbers of the Cartesian products of graphs
  106. Binomial convolution sum of divisor functions associated with Dirichlet character modulo 8
  107. Commutator of fractional integral with Lipschitz functions related to Schrödinger operator on local generalized mixed Morrey spaces
  108. System of degenerate parabolic p-Laplacian
  109. Stochastic stability and instability of rumor model
  110. Certain properties and characterizations of a novel family of bivariate 2D-q Hermite polynomials
  111. Stability of an additive-quadratic functional equation in modular spaces
  112. Monotonicity, convexity, and Maclaurin series expansion of Qi's normalized remainder of Maclaurin series expansion with relation to cosine
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  114. On the existence of tripartite graphs and n-partite graphs
  115. Classifying pentavalent symmetric graphs of order 12pq
  116. Almost periodic functions on time scales and their properties
  117. Some results on uniqueness and higher order difference equations
  118. Coding of hypersurfaces in Euclidean spaces by a constant vector
  119. Cycle integrals and rational period functions for Γ0+(2) and Γ0+(3)
  120. Degrees of (L, M)-fuzzy bornologies
  121. A matrix approach to determine optimal predictors in a constrained linear mixed model
  122. On ideals of affine semigroups and affine semigroups with maximal embedding dimension
  123. Solutions of linear control systems on Lie groups
  124. A uniqueness result for the fractional Schrödinger-Poisson system with strong singularity
  125. On prime spaces of neutrosophic extended triplet groups
  126. On a generalized Krasnoselskii fixed point theorem
  127. On the relation between one-sided duoness and commutators
  128. Non-homogeneous BVPs for second-order symmetric Hamiltonian systems
  129. Erratum
  130. Erratum to “Infinitesimals via Cauchy sequences: Refining the classical equivalence”
  131. Corrigendum
  132. Corrigendum to “Matrix stretching”
  133. Corrigendum to “A comprehensive review of the recent numerical methods for solving FPDEs”
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