Character codegrees in finite groups

Abstract

For an irreducible character \chi of a finite group G, we define its codegree as \mathrm{c}\mathrm{o}\mathrm{d}\left(\chi \right)=\frac{|G:\ker \chi |}{\chi \left(1\right)}. In this paper, we introduce some known results and unsolved problems about character codegrees in finite groups.

Keywords

Finite group / character / character codegree

1 Introduction

Throughout this paper, G always denotes a finite group, and the term “character” always refers to complex character unless otherwise stated. The standard notation and terminology are used, where \text{Irr}(G) and \mathrm{c}\mathrm{d}(G) represent the irreducible character set and the irreducible character degree set of G, respectively. For an irreducible character (or representation), the most important quantity is undoubtedly its degree. In the 1960s, Isaacs et al. initiated the theoretical research on character degrees, and since then, the study on character degrees has been a major topic in finite group theory with thousands of research results.

Twenty years ago, the author began investigating the dual arithmetic quantity of character degree, namely, character codegree. This topic has garnered increasing attention and is becoming a hot topic in the character theory of finite groups. In this paper, we present some known results and unsolved problems regarding the character codegrees of finite groups.

Definition 1.1 [25]　For \chi \in \text{Irr}(G), we define its codegree as

\mathrm{c}\mathrm{o}\mathrm{d}(\chi ):=\frac{|G:\ker \chi |}{\chi (1)}.

Let \mathrm{c}\mathrm{o}\mathrm{d}(G) be the set of codegrees of irreducible characters of G. For example, the degrees and codegrees of the five irreducible characters of the symmetry group {\mathrm{S}\mathrm{y}\mathrm{m}}_4 are as follows:

{\chi }_1(1)=1,\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_1)=1;{\chi }_2(1)=1,\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_2)=2;{\chi }_3(1)=2,\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_3)=3;

{\chi }_4(1)={\chi }_5(1)=3;\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_4)=\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_5)=8.

Note that the codegree of \chi \in \text{Irr}(G) is an integer factor of |G/\ker \chi | and |G|. Since

\chi (1)\cdot \mathrm{c}\mathrm{o}\mathrm{d}(\chi )=|G:\ker \chi |,

the character degree and the character codegree are dual arithmetic quantities with respect to the character. It follows from the equation |G|=\sum _{\chi \in \text{Irr}(G)}\chi (1{)}^2 that

\chi (1)⩽\mathrm{c}\mathrm{o}\mathrm{d}(\chi )

for all \chi \in \text{Irr}(G), with equality holding if and only if \chi =1_G. The following proposition is fundamental to character codegree and ensures that we can use induction in most cases.

Proposition 1.1 [25]　 Let \chi \in \text{Irr}(G).

(1) If N⊴G with N⩽\ker \chi, then \chi may be viewed as an irreducible character of G/N, and the codegrees of \chi in G and in G/N are the same.

(2) If M is subnormal in G and \psi is an irreducible constituent of {\chi }_M, then \mathrm{c}\mathrm{o}\mathrm{d}(\psi )\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ).

Proof　(1) This follows directly from the definition of character codegree.

(2) By induction we may assume that M is maximal normal in G. We note that \ker \chi \cap M⩽\ker \psi. Suppose first that \ker \chi ⩽M. Then \ker \chi ⩽\ker \psi, and since \chi (1)/\psi (1) divides |G/M|, we conclude

\mathrm{c}\mathrm{o}\mathrm{d}(\psi )=\frac{|M:\ker \psi |}{\psi (1)}|\frac{|G:\ker \chi |}{\chi (1)}=\mathrm{c}\mathrm{o}\mathrm{d}(\chi )

as desired. Suppose now that \ker \chi ⪇M. The maximality of M implies G=\ker \chi M, so {\chi }_M=\psi,

\mathrm{c}\mathrm{o}\mathrm{d}(\psi )=\frac{|M:\ker \psi |}{\psi (1)}=\frac{|M:M\cap \ker \chi |}{\psi (1)}=\frac{|G:\ker \chi |}{\chi (1)}=\mathrm{c}\mathrm{o}\mathrm{d}(\chi ),

and we are done.□

Note that the codegree of \chi \in \text{Irr}(G) is defined as |G|/\chi (1) in some literature, such as [8, 9, 34]. The main drawback of this definition is that the codegree of \chi in G is inconsistent with the codegree of \chi in G/\ker \chi.

What are the natural problems for the character codegrees? We believe that at least the following three aspects can be considered. Firstly, examine the arithmetic properties of the character codegrees themselves, though this area is almost unexplored. Secondly, study the relationship between character codegree and other arithmetic quantities, such as character degree and element order. Thirdly, investigate the dual problems of the character codegree corresponding to those of the character degree, which is a focus of most current research.

The fundamental Ito-Michler theorem determines which primes can divide a member in \mathrm{c}\mathrm{d}(G), while the analogous question for \mathrm{c}\mathrm{o}\mathrm{d}(G) is which primes can divide a member in \mathrm{c}\mathrm{o}\mathrm{d}(G). Let us answer the question below.

Proposition 1.2 [27]　 A prime p divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G) if and only if p\in \pi (G), that is, p is a prime divisor of |G|.

Proof　We only prove the necessity of the statement. Let L/E be a chief factor of G with p dividing |L/E|. Suppose that L/E is an elementary abelian p-group. Then p divides the codegree of any nonprincipal irreducible character of L/E. Suppose that L/E is a direct product of some isomorphic nonabelian simple groups. By Thompson’s theorem [15, Corollary 12.2], we can take a nonprincipal character \psi \in \text{Irr}(L/E) of p^{\prime }-degree. Therefore, in either case, there exists \psi \in \text{Irr}(L) with p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\psi ). Now let \chi be an irreducible constituent of {\psi }^G. Then Proposition 1.1 yields p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ), as wanted.□

We use the following notation in this paper: {\text{Irr}}^{\mathrm{♯}}(G) denotes the set of nonprincipal irreducible characters of G. For a character \psi, we use \text{Irr}(\psi ) to represent the set of irreducible constituents of \psi. If N⊴G, we write

\text{Irr}(G|N)=\{\chi \in \text{Irr}(G)\mid N\nleqq \ker \chi \},

\text{cd}(G|N)=\{\chi (1)\mid \chi \in \text{Irr}(G|N)\},

\text{cod}(G|N)=\{\text{cod}(\chi )\mid \chi \in \text{Irr}(G|N)\}.

Clearly, \text{Irr}(G|N)\cup \text{Irr}(G/N) is a partition of \text{Irr}(G). For a positive integer m and a prime p, let \pi (m) be the set of prime divisors of m, and use m_p or m{|}_p to represent the p-part of m.

2 Character codegree and element order

The arithmetic quantities of character codegree and element order arise from group representation and group structure, respectively. At first glance, there appears to be no necessary connection between these two quantities. However, our research has revealed a strong connection between them, which has motivated us to further investigate the character codegree.

Proposition 2.1 [27]　 Let G be a nonabelian simple group, and let p,q\in \pi (G) with p\ne q. Then pq divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Proof　Suppose pq does not divide any member in \mathrm{c}\mathrm{o}\mathrm{d}(G). Let a,b\in G be of order p and q respectively, and let \chi \in {\text{Irr}}^{\mathrm{♯}}(G). Then

\chi (1{)}_p=|G{|}_p\text{or}\chi (1{)}_q=|G{|}_q.

That is, \chi is of p-defect zero or q-defect zero. By [15, Theorem 8.17], we have \chi (a)\chi (b^{-1})=0. This leads to 0=\sum _{\chi \in \text{Irr}(G)}\chi (a)\chi (b^{-1})=1_G(a)1_G(b^{-1})=1, a contradiction.□

Let {\pi }_e(G) be the set of element orders of G.

Proposition 2.2 [27]　 Let p,q\in \pi (G) with p\ne q. If pq\in {\pi }_e(G), that is, G has an element of order pq, then pq divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Proof　Let E be a minimal normal subgroup of G. If \{p,q\}\subseteq \pi (E), then E is a direct product of nonabelian simple groups, and Proposition 2.1 shows that pq divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(E), thus Proposition 1.1 yields the required result. If pq\in {\pi }_e(G/E) (in particular, if E is a \{p,q{\}}^{\prime }-group), then induction and Proposition 1.1 also imply the result. Furthermore, if G has two different minimal normal subgroups E_1,E_2 satisfying p\in \pi (E_1) and q\in \pi (E_2), by Proposition 1.2, we can take \mu \in \text{Irr}(E_1) with p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\mu ) and \nu \in \text{Irr}(E_2) with q\mid \mathrm{c}\mathrm{o}\mathrm{d}(\nu ), so pq divides \mathrm{c}\mathrm{o}\mathrm{d}(\mu \times \nu ) and hence divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G). In summary, for any minimal normal subgroup E of G, we may assume that

p\in \pi (E),q\notin \pi (E),\text{and}pq\notin {\pi }_e(G/E).

Suppose that G has two different minimal normal subgroups E_1,E_2. Let x\in G be of order pq. The above assumptions yield x^q\in E_1\cap E_2=1, a contradiction. Hence, we may also assume that G has a unique minimal normal subgroup E.

Suppose the result is false and let x\in G with o(x)=pq. By the uniqueness of E, every \chi \in \text{Irr}(G|E) is faithful, so \chi (1{)}_p=|G{|}_p or \chi (1{)}_q=|G{|}_q. Obviously, x\in G\setminus E. Note that \chi (x)=0 for every \chi \in \text{Irr}(G|E) by [15, Theorem 8.17]. Now,

|C_G(x)|=\sum _{\chi \in \text{Irr}(G)}|\chi (x){|}^2=\sum _{\chi \in \text{Irr}(G/E)}|\chi (x){|}^2=|C_{G/E}(xE)|.

Since we assume pq\notin {\pi }_e(G/E), it follows pq\nmid |C_{G/E}(xE)|. However, pq obviously divides |C_G(x)|, a contradiction, and the proof is complete.□

Proposition 2.3 [28]　 Let G be a solvable group and let m be a square-free member in {\pi }_e(G). Then m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Furthermore, Isaacs removed the solvability assumption in Proposition 2.3 and proved the following nice result.

Theorem 2.1 [16]　 Let m be a square-free member in {\pi }_e(G). Then m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Proposition 2.4 [28]　 Let m\in {\pi }_e(G) be a prime power. Then m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Proposition 2.5　 Let G be an abelian group. Then {\pi }_e(G)=\mathrm{c}\mathrm{o}\mathrm{d}(G).

Proof　For an abelian group G, it is well known that \text{Irr}(G) forms a group and is isomorphic to G under the usual linear character multiplication. For each \lambda \in \text{Irr}(G), it is easy to see that \mathrm{c}\mathrm{o}\mathrm{d}(\lambda ) is exactly the order of the element \lambda in the abelian group \text{Irr}(G). Therefore, {\pi }_e(G)=\text{cod}(G).□

We have recently proven the following theorem, which is much stronger than Theorem 2.1 for solvable groups.

Theorem 2.2 [31]　 Let G be a solvable group and let m\in {\pi }_e(G). Then m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

In view of the above results, we have the following conjecture.

Conjecture 2.1　If m\in {\pi }_e(G), then m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

We think that Conjecture 2.1 is meaningful. Before fully solving this conjecture, there are several related issues that are worth investigating.

Problem 2.1　Suppose m\in {\pi }_e(G) with |\pi (m)|=2. Prove that m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Problem 2.2　Suppose m\in {\pi }_e(G) and G is a nonabelian simple group. Prove that m divides some member in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Problem 2.3　Let G be a finite group (or solvable group), and let \lambda be an irreducible character of some subgroup of G. Is there an irreducible component \chi of {\lambda }^G such that \mathrm{c}\mathrm{o}\mathrm{d}(\lambda )\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi )?

Note that if Problem 2.3 has an affirmative answer for all finite groups, then Conjecture 2.1 holds directly.

Proposition 2.3 can be restated as follows: If a solvable group G contains an element x, then there exists \chi \in \text{Irr}(G) such that \pi (o(x))\subseteq \pi (\mathrm{c}\mathrm{o}\mathrm{d}(\chi )). In [24], Moretó asked whether the converse of Proposition 2.3 holds, namely, If a solvable group G has an irreducible character \chi, is there an element x\in G such that \pi (\mathrm{c}\mathrm{o}\mathrm{d}(\chi ))\subseteq \pi (o(x))? Unfortunately, there are examples showing that the answer to this question is negative.

Conjecture 2.2　If m\in \mathrm{c}\mathrm{o}\mathrm{d}(G) and G is solvable, then there exists d\in {\pi }_e(G) such that |\pi (d)\cap \pi (m)|⩾\frac{1}{2}|\pi (m)|.

We have observed that, in addition to \mathrm{c}\mathrm{d}(G), {\pi }_e(G) can also be used as a frame of reference of \mathrm{c}\mathrm{o}\mathrm{d}(G).

3 The \mathit{p}-part of character codegree

Let p\in \pi (G), and let X be a set of positive integers associated with G, such as {\pi }_e(G), \mathrm{c}\mathrm{s}(G) (the set of conjugacy class sizes of G), \mathrm{c}\mathrm{d}(G) and \mathrm{c}\mathrm{o}\mathrm{d}(G). For any member m of such X, we have 1⩽m_p⩽|G{|}_p. Typically, we study the p-part of X by estimating the largest or smallest p-part of its members and by giving a structural description of the group G under the assumption that the p-parts of the members in X are all “small” or all “large”. We define

e_p(G)=max\{{\log }_p(\chi (1{)}_p)|\chi \in \text{Irr}(G)\},

c_p(G)=max\{{\log }_p(\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p)|\chi \in \text{Irr}(G)\}.

There are many results on the p-part of character degree (see [30, 41] and their references), including the well-known Ito-Michler theorem which states that e_p(G)=0 if and only if G has a normal abelian Sylow p-subgroup. As to c_p(G), the following one is fundamental.

Theorem 3.1 [27]　 For a prime p\in \pi (G), we have

(1) c_p(G)⩾1;

(2) for a nonidentity normal subgroup N of G, p does not divide any member in \mathrm{c}\mathrm{o}\mathrm{d}(G|N) if and only if P\in {\mathrm{S}\mathrm{y}\mathrm{l}}_p(G) acts fixed-point-freely on N, that is, PN is a Frobenius group with P as its complement and N as its kernel.

Proof　(1) This follows by Proposition 1.2.

(2) We only need to prove the necessity. We claim that p\notin \pi (N). Otherwise, by Proposition 1.2, we may take \lambda \in {\text{Irr}}^{\mathrm{♯}}(N) of codegree divisible by p. Then, for any \chi \in \text{Irr}({\lambda }^G), we have \chi \in \text{Irr}(G|N) and p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ), which leads to a contradiction. Thus p\notin \pi (N) as claimed, so P\in {\mathrm{S}\mathrm{y}\mathrm{l}}_p(G) acts coprimely on N. Now it suffices to show that P acts fixed-point-freely on N. Let E be a minimal normal subgroup of G.

Suppose p\in \pi (E). Let \mu \in \text{Irr}(E) with p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\mu ), and let \nu \in {\text{Irr}}^{\mathrm{♯}}(N). Note that EN=E\times N, and it follows that \mu \nu \in \text{Irr}(EN|N) with p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\mu \nu ). For any \chi \in \text{Irr}((\mu \nu {)}^G), we have \chi \in \text{Irr}(G|N) and p\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ), a contradiction. Therefore, all minimal normal subgroups E of G are p^{\prime }-groups. By replacing (G,N) with (G/E,NE/E) and noting that the condition is still valid for (G/E,NE/E), we conclude by induction that either PE/E acts fixed-point-freely on NE/E or NE=E.

Suppose that G has two different minimal normal subgroups E_1,E_2. Since P{E}_i/E_i acts fixed-point-freely on N{E}_i/E_i (or N=E_i) for i=1,2, it follows easily that P also acts fixed-point-freely on N, and we are done.

Suppose that G has a unique minimal normal subgroup E. Then E⩽N and \text{Irr}(G|E)\subseteq \text{Irr}(G|N). For every \chi \in \text{Irr}(G|E), since \chi is faithful and of codegree not divisible by p, we have \chi (1{)}_p=|P|. This implies that {\psi }^{PN} is irreducible for every \psi \in \text{Irr}(N|E). For \psi \in {\text{Irr}}^{\mathrm{♯}}(N/E), since PE/E acts fixed-point-freely on N/E, {\psi }^{PN} is also irreducible. Therefore, every \psi \in {\text{Irr}}^{\mathrm{♯}}(N) induces an irreducible character of PN, which forces PN to be a Frobenius group with kernel N and complement P.□

Regarding the p-part of the character codegree, there are two independent works [5] and [33]. It is proved in [5] that if G is p-solvable, then the p-length of G is at most c_p(G). However, [33] provides a stronger result as stated in the following theorem.

Theorem 3.2 [33]　 For a finite group G and a prime p, there exists a normal subgroup N of G such that O_{p^{\prime }}(G)⩽N⩽O_{p^{\prime }p}(G), O_{p^{\prime }p}(G/N)=O_{p^{\prime }p}(G)/N, and G/N possesses a faithful irreducible character \chi of p^{\prime }-degree. In particular, we have

{\log }_p(|G/O_{p^{\prime }p}(G){|}_p)⩽{\log }_p(|G/N{|}_p)⩽c_p(G).

Corollary 3.1 [33]　 Let G be a finite group and p be a prime. The following results hold:

(1) if O_{p^{\prime }p}(G)=O_{p^{\prime }}(G), then {\log }_p(|G{|}_p)=c_p(G);

(2) if G is p-solvable, then .

The previous results show that c_p(G) is “large”, prompting the natural question of whether e_p(G) (or {\log }_p(|G/O_p(G){|}_p)) can be bounded in terms of c_p(G). In other words, does there exist a function f such that e_p(G)⩽f(c_p(G)) for all finite groups G? Remark 3.1(1) below shows that the answer to this question is negative, and Remark 3.1(2) indicates that c_p(G) cannot be bounded by e_p(G).

Remark 3.1　(1) Let {\mathbb{Z}}_p act faithfully and irreducibly on an elementary abelian q-group V, where p and q are different primes. Let G be the direct product of n copies of {\mathbb{Z}}_p⋉V. Then, it is easy to verify that

{\log }_p(|G/O_p(G){|}_p)=n,e_p(G)=nand{c}_p(G)=1.

Since n can be arbitrarily large, neither {\log }_p(|G/O_p(G){|}_p) nor e_p(G) can be bounded by c_p(G).

(2) Let a p^{\prime }-group A act faithfully and irreducibly on an elementary abelian p-group V of order p^n, and let G=A⋉V. Then, it is easy to verify that e_p(G)=0 and c_p(G)=n. This shows that c_p(G) cannot be bounded by e_p(G).

The part (1) of Theorem 3.1 indicates that c_p(G)=0 if and only if p\nmid |G|. Now we consider the case when c_p(G)=1, which means p^2 does not divide any irreducible character codegree of G. Since all groups G with |G{|}_p=p satisfy c_p(G)=1, we only need to consider the case where |G{|}_p⩾p^2. It is shown in [5] that if c_p(G)=1 and |G{|}_p⩾p^2, then G is p-solvable with an elementary abelian Sylow p-subgroup.

Theorem 3.3 [33]　 Let p be a prime and G be a finite group with |G{|}_p⩾p^2. Then, c_p(G)=1 if and only if G has a normal series G⩾O^{p^{\prime }}(G)=P⋉V>V⩾1 satisfying the following conditions:

(1) P\in {\mathrm{S}\mathrm{y}\mathrm{l}}_p(G) is elementary abelian, V=O^{p^{\prime }p}(G) is a p^{\prime }-group;

(2) viewing P as a linear space over {\mathbb{F}}_p, we have H/C_H(P)⩽Z(\mathrm{G}\mathrm{L}(P)), where H\in {\mathrm{H}\mathrm{a}\mathrm{l}\mathrm{l}}_{p^{\prime }}(G/V);

(3) V is solvable and C_V(P)=1;

(4) if in addition |P|⩾p^3, then V is nilpotent, and V=C_V(a)⋉[V,⟨a⟩] for all a\in P.

Let p\in \pi (G) and assume that p\nmid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for any nonlinear \chi \in \text{Irr}(G). By Theorem 3.1, we see that either G is abelian or P\in {\mathrm{S}\mathrm{y}\mathrm{l}}_p(G) acts fixed-point-freely on G^{\prime }, and in particular the Fitting height of G is at most 2.

Problem 3.1　Let p\in \pi (G) and assume that p^2\nmid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for any nonlinear \chi \in \text{Irr}(G). Describe the structure of G.

Problem 3.2　For a given prime p and for every irreducible p-Brauer character of G, we can similarly define its codegree. Investigate the structure of G in which p or p^2 does not divide any irreducible p-Brauer character codegree of G.

Proposition 3.1 [4]　 Let p\in \pi (G) and e be a positive integer. If m_p=p^e for all , then e=1 and G is an elementary abelian p-group.

Problem 3.3　Let p be a prime and e be a positive integer. Study the structure of G in which m_p\in \{1,p^e\} for all m\in \mathrm{c}\mathrm{o}\mathrm{d}(G).

Problem 3.4　Describe the structure of nonsolvable groups G in which c_2(G) is small; for related work, see [26].

4 The codegree graph

Let X be a set of positive integers associated with a finite group G, such as {\pi }_e(G), \mathrm{c}\mathrm{s}(G), \mathrm{c}\mathrm{d}(G) and \mathrm{c}\mathrm{o}\mathrm{d}(G). For the convenience of expressing and studying the relationship between the members of X, we introduce an undirected simple graph \mathrm{\Gamma }(X) as follows: the vertices are prime divisors of members of X, and two distinct vertices p and q are connected by an edge if and only if pq divides some member of X.

The graph \mathrm{\Gamma }(X) has been extensively studied when X={\pi }_e(G),\mathrm{c}\mathrm{s}(G) or \mathrm{c}\mathrm{d}(G), and the research content is roughly as follows.

(1) Describing the graph-theoretical properties of \mathrm{\Gamma }(X) for finite groups in general or for some large class of finite groups.

(2) Describing the group structure of G under some graph-theoretical assumption of \mathrm{\Gamma }(X). Notice that existing results have shown that \mathrm{\Gamma }(X) usually has strong connectivity. This indicates that only when the connectivity of \mathrm{\Gamma }(X) is assumed to be not too strong (at least not a complete graph), it is possible to obtain a description of the structure of the group.

We use \mathrm{\Gamma }(G) to denote the codegree graph \mathrm{\Gamma }(\mathrm{c}\mathrm{o}\mathrm{d}(G)) of G. We know that \pi (G) is the set of vertices of \mathrm{\Gamma }(G) by Propositions 1.2, and that \mathrm{\Gamma }(G) is a complete graph when G is a simple group by Proposition 2.1. Furthermore, Proposition 2.2 implies that \mathrm{\Gamma }({\pi }_e(G)), often called the prime graph of G, is a subgraph of \mathrm{\Gamma }(G), and the two graphs have the same set of vertices. This indicates that the codegree graph of a group has stronger connectivity than the prime graph of the group. The following theorem presents the basic graph-theoretic properties of the codegree graph.

Theorem 4.1 [27]　 For distinct primes p,q,r in \pi (G), there exist distinct u and v in \{p,q,r\} such that uv divides some member of \mathrm{c}\mathrm{o}\mathrm{d}(G), meaning that there is at least one edge in the codegree graph \mathrm{\Gamma }(G) connecting two points from p,q, and r. In particular,

(1) \mathrm{\Gamma }(G) has at most two connected components;

(2) if \mathrm{\Gamma }(G) is connected, \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{\Gamma }(G)), the diameter of \mathrm{\Gamma }(G), is at most 3.

We now discuss the connection between the graph-theoretical properties of \mathrm{\Gamma }(G) and the structural properties of G, starting with the two classes of graphs with the least connectivity: triangle-free graphs and disconnected graphs.

Proposition 4.1 [3]　 If \mathrm{\Gamma }(G) does not contain triangles, then G is solvable and |\pi (G)|⩽5.

Problem 4.1　Classify finite groups G with triangle-free \mathrm{\Gamma }(G).

If all character codegrees of G are prime powers, then the codegree graph \mathrm{\Gamma }(G) is disconnected unless G is of prime power order. Finite groups whose character codegrees are all or almost all primes have been studied in [25] and [3]; moreover, finite groups with all character codegrees being prime powers have been classified in [27] and [37].

Recall that a finite group G with normal subgroups M and N such that G/N is a Frobenius group with kernel M/N and M is a Frobenius group with kernel N is called a 2-Frobenius group. For finite groups with disconnected codegree graphs, we have the following theorem.

Theorem 4.2 [27]　 The codegree graph \mathrm{\Gamma }(G) is disconnected if and only if G is a Frobenius or 2-Frobenius group, and in this case, \mathrm{\Gamma }(G) has exactly two connected components, each of which is a complete graph.

Let , and let \mathrm{\Gamma }(G|N) be the graph defined by the set \mathrm{c}\mathrm{o}\mathrm{d}(G|N). Clearly, \mathrm{\Gamma }(G|N) is a subgraph of \mathrm{\Gamma }(G). It is proved in [27] that \mathrm{\Gamma }(G|N) has at most two connected components; if \mathrm{\Gamma }(G|N) is disconnected and G is nonsolvable, then G is very close to being a Frobenius group; if \mathrm{\Gamma }(G|N) is disconnected and G is solvable, then the Fitting height of G is at most 4.

We note that the diameter of codegree graph of a finite group can reach the upper bound 3. For example, consider G=({\mathbb{Z}}_2\times {\mathbb{Z}}_3)⋉({\mathbb{Z}}_5\times {\mathbb{Z}}_7) satisfying the following conditions: {\mathbb{Z}}_2 acts fixed-point-freely on {\mathbb{Z}}_5\times {\mathbb{Z}}_7, while {\mathbb{Z}}_3 acts trivially on {\mathbb{Z}}_5 and fixed-point-freely on {\mathbb{Z}}_7. Then \mathrm{c}\mathrm{o}\mathrm{d}(G)=\{2,3,6,35,5,15\}, the distance between 2 and 7 in \mathrm{\Gamma }(G) is equal to 3, and so \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{\Gamma }(G))=3.

Problem 4.2　Classify the finite groups G with \mathrm{d}\mathrm{i}\mathrm{a}\mathrm{m}(\mathrm{\Gamma }(G))=3.

Problem 4.3　Given p,q\in \pi (G) such that p and q are not connected by an edge in the codegree graph \mathrm{\Gamma }(G), investigate the \{p,q\}-structure of G.

Theorem 4.3 [1]　 If F(G)=1, where F(G) is the Fitting subgroup of G, then \mathrm{\Gamma }(G) is a complete graph.

The above theorem generalizes Proposition 2.1, but its proof relies on the classification of finite simple groups.

Problem 4.4　Can Theorem 4.3 be proven without using the classification of finite simple groups?

As previously pointed out, the prime graph of a finite group must be a subgraph of its codegree graph. Moreover, there exist examples showing that the prime graph of a solvable group can be a proper subgraph of its codegree graph.

Problem 4.5　Provide a characteristic description of finite solvable groups for which the prime graph and codegree graph differ.

Problem 4.6　Describe the common properties of prime graphs and codegree graphs of solvable groups.

Conjecture 4.1　For an undirected simple graph \mathrm{\Gamma }, \mathrm{\Gamma } is the prime graph of some solvable group G_1 if and only if it is the codegree graph of some solvable group G_2.

5 Character codegree and character degree

In this section, we describe the group structure by comparing the size relationship, divisibility relationship, or coprimelity relationship between character degrees and character codegrees of the group. For a nonprincipal irreducible character \chi, \chi (1) must be strictly less than \mathrm{c}\mathrm{o}\mathrm{d}(\chi ). However, it is possible for a prime p that \chi (1{)}_p>\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p or .

Theorem 5.1 [12]　 A finite group G is nilpotent if and only if \chi (1)\mid \mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for all \chi \in \text{Irr}(G).

Localizing the above theorem, we obtain the following Theorem 5.2, which is a generalization of the Ito-Michler theorem.

Theorem 5.2 [29]　 Let p be a prime and assume that \chi (1{)}_p⩽\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p for all \chi \in \text{Irr}(G). Then one of the following holds:

(1) G is p-closed, i.e., G has a normal Sylow p-subgroup;

(2) p=3, and G has a composition factor isomorphic to the alternating group {\mathrm{A}\mathrm{l}\mathrm{t}}_7;

(3) p=2, and G has a composition factor isomorphic to {\mathrm{A}\mathrm{l}\mathrm{t}}_7, {\mathrm{A}\mathrm{l}\mathrm{t}}_{11}, or Mathieu group {\mathrm{M}}_{22}.

Corollary 5.1 [29]　 Let p be a prime and assume that either p⩾3 or G is p-solvable. Then G is p-closed if and only if for all \chi \in \text{Irr}(G) satisfying p\mid \chi (1).

Corollary 5.2 [29]　 For a prime p⩾5, the following statements are equivalent:

(1) G is p-closed;

(2) for all \chi \in \text{Irr}(G) satisfying p\mid \chi (1);

(3) \chi (1{)}_p⩽\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p for all \chi \in \text{Irr}(G).

The following result (Theorem 5.3) due to Tong-Viet is actually the Brauer character version of Theorem 5.2.

Theorem 5.3 [36]　 Let p be a prime and assume that [\varphi (1{)}_p{]}^{{\mu }_p}⩽\mathrm{c}\mathrm{o}\mathrm{d}(\varphi ){|}_p for all irreducible p-Brauer characters \varphi of G, where {\mu }_2=8, {\mu }_3=2, and {\mu }_p=1 for p⩾5. Then G is p-closed.

Problem 5.1　Study the non-p-solvable groups G in which [\chi (1{)}_p{]}^2⩽\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p for every \chi \in \text{Irr}(G), or \chi (1{)}_p⩽[\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){]}^2 for every \chi \in \text{Irr}(G), where p is a given prime.

Problem 5.2　Suppose that \pi (\chi (1))\subseteq \pi (\mathrm{c}\mathrm{o}\mathrm{d}(\chi )) for every irreducible character \chi of G. Is G solvable? If so, describe the structure of G.

We introduce some sufficient conditions for the solvability of finite groups by examining the size relationship between character degrees and character codegrees. The author once conjectured that if \chi (1{)}^2⩾\mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for all nonlinear \chi \in \text{Irr}(G), then G is solvable. Although the conjecture is not true, Gao and Liu obtained the following result.

Proposition 5.1 [13]　 If \chi (1{)}^e>\mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for all nonlinear \chi \in \text{Irr}(G), where

e=\frac{\log (460815505920)}{\log (10944)}-1\approx 1.8876,

then G is solvable.

Problem 5.3　Study the finite groups G in which \chi (1{)}^2⩾\mathrm{c}\mathrm{o}\mathrm{d}(\chi ) for all nonlinear \chi \in \text{Irr}(G). Given the classification for the nonsolvable case, estimate the upper bound of the derived length and the Fitting height of G for the solvable case.

Proposition 5.2 [38]　 If \mathrm{c}\mathrm{o}\mathrm{d}(\chi )⩽p(\chi )\cdot \chi (1) for all nonlinear \chi \in \text{Irr}(G), where p(\chi ) is the largest prime divisor of |G:\ker \chi |, then G is solvable.

Problem 5.4　Describe the structure of the group in Proposition 5.2.

Conjecture 5.1　Suppose that for all nonlinear \chi \in \text{Irr}(G), where d(\chi ) is the order of the largest Sylow subgroup of |G/\ker \chi |. Then G is solvable (If this is not true, classify the nonsolvable groups satisfying the assumption).

A finite group G is called an \mathcal{H}-group if \mathrm{g}\mathrm{c}\mathrm{d}(\chi (1),\mathrm{c}\mathrm{o}\mathrm{d}(\chi ))=1 for every \chi \in \text{Irr}(G). In [19], we gave a complete description of all \mathcal{H}-groups (see also [20]), here we only describe the nonsolvable \mathcal{H}-groups.

Theorem 5.4 [19]　 A nonsolvable group G is an \mathcal{H}-group if and only if it satisfies the following conditions:

(1) G=D⋉(L\times N), where D, L and N are Hall subgroups of G;

(2) L=L^{\prime }, L has a solvable normal subgroup E such that L/E\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,2^f), where f⩾2;

(3) E=1, or |E|=2^{2f} and \mathrm{c}\mathrm{d}(L|E)=2^{2f}-1;

(4) D is cyclic, |D| is a square-free divisor of f;

(5) G/L\cong D⋉N is a solvable \mathcal{H}-group.

Problem 5.5　For a given prime p, study the p-structure of finite groups G in which p does not divide \mathrm{g}\mathrm{c}\mathrm{d}(\chi (1),\mathrm{c}\mathrm{o}\mathrm{d}(\chi )) for any \chi \in \text{Irr}(G).

Problem 5.6　Let p be a prime and k be a positive integer. Study the p-solvable groups G in which p^{k+1} does not divide \mathrm{g}\mathrm{c}\mathrm{d}(\chi (1),\mathrm{c}\mathrm{o}\mathrm{d}(\chi )) for any \chi \in \text{Irr}(G).

Problem 5.7　Study the finite groups G in which \mathrm{g}\mathrm{c}\mathrm{d}(\chi (1),\mathrm{c}\mathrm{o}\mathrm{d}(\chi )) is square-free for all \chi \in \text{Irr}(G). If G is solvable, estimate the upper bounds for the Fitting height of G and the rank of G/F(G); if G is nonsolvable, provide a complete description of it.

Problem 5.8　Study the finite groups G in which \chi (1{)}_p⩽\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p for every \chi \in \text{Irr}(G) and every prime divisor p of \mathrm{c}\mathrm{o}\mathrm{d}(\chi ).

Problem 5.9　Study the finite groups G in which \chi (1{)}_p⩾\mathrm{c}\mathrm{o}\mathrm{d}(\chi ){|}_p for every \chi \in \text{Irr}(G) and every prime divisor p of \chi (1).

Obviously, each \mathcal{H}-group satisfies the conditions given in above two problems.

Problem 5.10　If G is solvable and \mathrm{g}\mathrm{c}\mathrm{d}(\chi (1),\mathrm{c}\mathrm{o}\mathrm{d}(\chi ))>1 for all nonlinear \chi \in \text{Irr}(G), is there an upper bound for the Fitting height of G?

6 \mathit{\rho }-\mathit{\sigma } conjuncture

Let X be a set of positive integers associated with a finite group G, such as {\pi }_e(G), \mathrm{c}\mathrm{s}(G), \mathrm{c}\mathrm{d}(G) and \mathrm{c}\mathrm{o}\mathrm{d}(G). We define \rho (X) as the set of primes that divide some member of X, and let

\sigma (X)=max\{|\pi (m)|:m\in X\}.

The \rho-\sigma conjecture for \mathrm{c}\mathrm{d}(G) was proposed by Huppert, and it is one of the central problems in character theory. The \rho-\sigma conjecture for general X is as follows: there exists a constant c such that |\rho (X)|⩽c\cdot \sigma (X) for all finite groups. Obviously,

(1)\begin{array}{r}\rho ({\pi }_e(G))=\rho (\mathrm{c}\mathrm{o}\mathrm{d}(G))=\pi (G).\end{array}

For simplicity, we write

(2)\begin{array}{r}{\sigma }_e(G)=\sigma ({\pi }_e(G)),{\sigma }_{co}(G)=\sigma (\mathrm{c}\mathrm{o}\mathrm{d}(G)).\end{array}

Note that by Isaacs’ result (Theorem 2.1), we have

(3)\begin{array}{r}{\sigma }_e(G)⩽{\sigma }_{co}(G).\end{array}

For the \rho-\sigma conjecture on element orders, Keller proved the following well-known result for solvable groups.

Theorem 6.1 [18]　 For every solvable group G, we have

|\pi (G)|⩽C(|\pi (G)|)\cdot {\sigma }_e(G),

where C(n) is a function satisfying \underset{n\to \mathrm{\infty }}{lim}C(n)=4. In particular, there exists a constant C such that |\pi (G)|⩽C\cdot {\sigma }_e(G) for every solvable group G.

Using equalities (1), (2), inequality (3), and Keller's theorem, we immediately get the affirmative answer to the \rho-\sigma conjecture on character codegrees for solvable groups, and then extend it to all finite groups.

Theorem 6.2 [40]　 There exists a constant c such that |\pi (G)|⩽c\cdot {\sigma }_{co}(G) for all finite groups G.

Yang [39] improved the constant c in Theorem 6.2 to an absolute constant \frac{23}{3}. For the solvable case, we directly applied Keller's result without essential study on {\sigma }_{co}(G). Therefore, the absolute constant \frac{23}{3} still has room for improvement.

Conjecture 6.1　For every solvable group G, we have |\pi (G)|⩽3{\sigma }_{co}(G).

By Theorem 6.3(3) below, the absolute constant 3 in Conjecture 6.1 cannot be improved.

Conjecture 6.2　For every finite group G, we have |\pi (G)|⩽4{\sigma }_{co}(G).

Theorem 6.3 [24, 17, 42]　 For solvable groups G, the following results hold:

(1) if {\sigma }_e(G)=2 or {\sigma }_{co}(G)=2, then the exact upper bound of |\pi (G)| is 5;

(2) if {\sigma }_e(G)=3 or {\sigma }_{co}(G)=3, then the exact upper bound of |\pi (G)| is 8;

(3) if {\sigma }_e(G)=4 or {\sigma }_{co}(G)=4, then the exact upper bound of |\pi (G)| is 12.

Problem 6.1　Consider only solvable groups G. For a positive integer n, let f_1(n) denote the exactly upper bound of |\pi (G)| when {\sigma }_e(G)=n, and let f_2(n) denote the exact upper bound of |\pi (G)| when {\sigma }_{co}(G)=n. Is there a positive integer m such that f_1(m)\ne f_2(m)?

Problem 6.2　Study finite groups G with {\sigma }_{co}(G)=2. Note that some special cases are discussed in [35].

Suppose that G is a solvable group. It is easy to prove that {\sigma }_e(G)=1 if and only if {\sigma }_{co}(G)=1. However, we do not know whether {\sigma }_e(G)=2 if and only if {\sigma }_{co}(G)=2? There are some more questions worth discussing in this direction.

Problem 6.3　Study the finite groups in which there is no prime dividing three character codegrees. See [32] and its references for the dual problem of the degree version.

Problem 6.4　Let k be a positive integer and suppose that there is no prime dividing k members in \mathrm{c}\mathrm{o}\mathrm{d}(G). Is |\pi (G)| or |\mathrm{c}\mathrm{o}\mathrm{d}(\mathrm{G})| bounded by some function of k?

7 Others

In this section, we continue to present our results and problems regarding character codegrees. Note that these results and problems also have corresponding versions with respect to character degrees as shown in [14, 15].

We start by discussing the nilpotent class of p-groups. In [11], an upper bound for the nilpotent class of a p-group in terms of its character codegrees was established, and the character codegrees of a maximal class p-group were investigated in [10]. For a dihedral group G of order 2^n, its nilpotent class is n-1, and \mathrm{c}\mathrm{d}(G)=\{1,2\}, so e_2(G)=1. This implies that the nilpotent class of a p-group cannot be bounded by e_p(G). Interestingly, however, the nilpotent class of a p-group can be bounded by c_p(G).

Theorem 7.1 [11]　 Let G be a p-group with c_p(G)=a⩾2. Then the nilpotent class of G does not exceed 2a-2. Furthermore, if p=2 and a⩾3, or if p=3 and a⩾4, then the nilpotent class of G does not exceed 2a-3.

Problem 7.1　Is the nilpotent class of a p-group G bounded by |\mathrm{c}\mathrm{o}\mathrm{d}(G)|?

Problem 7.2　Is the nilpotent class of a Sylow p-subgroup of a finite group G bounded by c_p(G)?

Next, we discuss the relationship between |\mathrm{c}\mathrm{o}\mathrm{d}(G)| and the structure of G. Obviously |\mathrm{c}\mathrm{o}\mathrm{d}(G)|=1 if and only if G=1; and if |\mathrm{c}\mathrm{o}\mathrm{d}(G)|=2, then it is easy to verify that G is an elementary abelian p-group for some prime p.

Proposition 7.1 [2]　 If |\mathrm{c}\mathrm{o}\mathrm{d}(G)|=3, then either G is a p-group of class 2, or |\pi (G)|=2, and G is a Frobenius group with a complement of prime order.

The following proposition describes the nonsolvable groups with few character codegrees. Note that Proposition 7.1 implies that |\mathrm{c}\mathrm{o}\mathrm{d}(G)|⩾4 for all nonsolvable groups G.

Proposition 7.2 [22, 23]　 For a nonsolvable group G, the following hold.

(1) If |\mathrm{c}\mathrm{o}\mathrm{d}(G)|=4, then G\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,2^f) for some integer f⩾2.

(2) If |\mathrm{c}\mathrm{o}\mathrm{d}(G)|=5, then one of the following holds:

　　 1) G has a minimal normal subgroup M of order 2^{2f}, with exactly two G-conjugacy classes, and G/M\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,2^f);

　 2) G\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,q), where q⩾7 is an odd prime power;

　 3) G\cong \mathrm{P}\mathrm{G}\mathrm{L}(2,q), where q⩾5 is an odd prime power;

　 4) G\cong {\mathrm{M}}_{10}.

Conjecture 7.1　The Fitting height of a solvable group G does not exceed |\mathrm{c}\mathrm{o}\mathrm{d}(G)|.

Conjecture 7.2　The derived length of a solvable group G does not exceed |\mathrm{c}\mathrm{o}\mathrm{d}(G)|.

Since there is currently no efficient method to estimate the number of character codegrees of a finite group, Conjecture 7.2 seems difficult to verify even for p-groups. It is unclear what kind of number |\mathrm{c}\mathrm{o}\mathrm{d}(G)| is.

Problem 7.3　Is there any relationship between |\mathrm{c}\mathrm{d}(G)| and |\mathrm{c}\mathrm{o}\mathrm{d}(G)|?

Problem 7.4　Study the finite groups G in which {\chi }_1(1)={\chi }_2(1) if and only if \mathrm{c}\mathrm{o}\mathrm{d}({\chi }_1)=\mathrm{c}\mathrm{o}\mathrm{d}({\chi }_2) for every pair of nonlinear irreducible characters {\chi }_1,{\chi }_2 of G.

Now we discuss how \mathrm{c}\mathrm{o}\mathrm{d}(G) determines the structure of G. A well-known conjecture proposed by Huppert states: If \mathrm{c}\mathrm{d}(G)=\mathrm{c}\mathrm{d}(S), where G is a finite group and S is a nonabelian simple group, then G\cong S\times A for some abelian group A. Similarly, the author proposed the dual conjecture as follows.

Conjecture 7.3　Let G be a finite group and S be a nonabelian simple group. If \mathrm{c}\mathrm{o}\mathrm{d}(G)=\mathrm{c}\mathrm{o}\mathrm{d}(S), then G\cong S.

It is proved in [6], the first published article on the conjecture, that Conjecture 7.3 holds when S\cong \mathrm{P}\mathrm{S}\mathrm{L}(2,q).

Problem 7.5　Let H be a finite group and G a solvable group. If \mathrm{c}\mathrm{o}\mathrm{d}(G)=\mathrm{c}\mathrm{o}\mathrm{d}(H), is H also solvable? Alternatively, is there a solvable section of H whose order is close to that of |G|?

We acknowledge that Problem 7.5 may be challenging. However, we propose a more specific conjecture worth studying.

Conjecture 7.4　Let H be a finite group and G an abelian group. If \mathrm{c}\mathrm{o}\mathrm{d}(G)=\mathrm{c}\mathrm{o}\mathrm{d}(H), then H is solvable.

Next, we introduce the dual object of complex group algebra. For a given m\in \mathrm{c}\mathrm{o}\mathrm{d}(G), we define n_G(m) as the number of irreducible characters \chi \in \text{Irr}(G) with codegree m, and we call it the multiplicity of m. Set \mathrm{C}\mathrm{C}(G)=\{(m,n_G(m))|m\in \mathrm{c}\mathrm{o}\mathrm{d}(G)\}. Although \mathrm{C}\mathrm{C}(G) lacks an algebraic structure, it is really dual to the complex group algebra \mathbb{C}(G) of G.

We know that the structure of G is almost determined by its group algebra \mathbb{C}(G), now it's natural to inquire about the extent to which \mathrm{C}\mathrm{C}(G) can determine the structure of G. Unfortunately, there is currently no method to calculate the multiplicity of a codegree, and this also leads us to know almost nothing about \mathrm{C}\mathrm{C}(G). Here we list only two questions regarding \mathrm{C}\mathrm{C}(G).

Problem 7.6　Let G_1 and G_2 be two finite groups with \mathrm{C}\mathrm{C}(G_1)=\mathrm{C}\mathrm{C}(G_2). Does it follow that |G_1|=|G_2|?

Problem 7.7　Let G_1 and G_2 be two finite groups with \mathrm{C}\mathrm{C}(G_1)=\mathrm{C}\mathrm{C}(G_2). If G_1 is abelian, does it follow that G_2 is also abelian?

It is worth noting that examining the average codegree \frac{1}{|\text{Irr}(G)|}\sum _{\chi \in \text{Irr}(G)}\mathrm{c}\mathrm{o}\mathrm{d}(\chi ) of G may yield interesting results. Although we only discuss the codegrees for irreducible complex characters here, we believe that there are parallel results or questions for irreducible Brauer characters in many situations. Additionally, investigating the codegrees for only a subset of the irreducible characters can also lead to outcomes, as demonstrated in [21, Theorem 1.4] and [7].

References

[1] Ahanjideh N. The Fitting subgroup, p-length, derived length and character table. Math Nachr 2021; 294(2): 214–223

[2] Alizadeh F, Behravesh H, Ghaffarzadeh M, Ghasemi M, Hekmatara S. Groups with few codegrees of irreducible characters. Comm Algebra 2019; 47(3): 1147–1152

[3] Alizadeh F, Ghasemi M, Ghaffarzadeh M. Finite groups whose codegrees are almost prime. Comm Algebra 2021; 49(2): 538–544

[4] Bahramian R, Ahanjideh N. p-divisibility of co-degrees of irreducible characters. Bull Aust Math Soc 2021; 103(1): 78–82

[5] Bahramian R, Ahanjideh N. p-parts of co-degrees of irreducible characters. Comptes Rendus Math Acad Sci Paris 2021; 359: 79–83

[6] Bahri A, Akhlaghi Z, Khosravi B. An analogue of Huppert’s conjecture for character codegrees. Bull Aust Math Soc 2021; 104(2): 278–286

[7] Chen X, Yang Y. Normal p-complements and monomial characters. Monatsh Math 2020; 193(4): 807–810

[8] Chillag D, Herzog M. On character degree quotients. Arch Math (Basel) 1990; 55(1): 25–29

[9] Chillag D, Mann A, Manz O. The codegrees of irreducible characters. Israel J Math 1991; 73(2): 207–223

[10] Croome S, Lewis M L. Character codegrees of maximal class p-groups. Canad Math Bull 2020; 63(2): 328–334

[11] Du N, Lewis M L. Codegrees and nilpotence class of p-groups. J Group Theory 2016; 19(4): 561–567

[12] Gagola S M Jr, Lewis M L. A character theoretic condition characterizing nilpotent groups. Comm Algebra 1999; 27(3): 1053–1056

[13] Gao Y, Liu Y. On codegrees and solvable groups. Bull Iranian Math Soc 2022; 48(4): 1357–1363

[14] HuppertB. Character Theory of Finite Groups. De Gruyter Expositions in Mathematics, Vol 25. Berlin: Walter de Gruyter & Co, 1998

[15] IsaacsI M. Character Theory of Finite Groups. NewYork: Academic Press, 1976

[16] Isaacs I M. Element orders and character codegrees. Arch Math (Basel) 2011; 97(6): 499–501

[17] Keller T M. Solvable groups with a small number of prime divisors in the element orders. J Algebra 1994; 170(2): 625–648

[18] Keller T M. A linear bound for ρ(n). J Algebra 1995; 178(2): 643–652

[19] Liang D, Qian G. Finite groups with coprime character degrees and codegrees. J Group Theory 2016; 19(5): 763–776

[20] Liang D, Qian G, Shi W. Finite groups whose all irreducible character degrees are Hall-numbers. J Algebra 2007; 307(2): 695–703

[21] Liu J, Wu K, Meng W. P-characters and the structure of finite solvable groups. J Group Theory 2021; 24(2): 285–291

[22] Liu Y, Yang Y. Nonsolvable groups with four character codegrees. J Algebra Appl 2021; 20(11): Paper No 2150196 (5 pp)

[23] Liu Y, Yang Y. Nonsolvable groups with five character codegrees. Comm Algebra 2021; 49(3): 1274–1279

[24] Moretó A. Huppert’s conjecture for character codegrees. Math Nachr 2021; 294(11): 2232–2236

[25] Qian G. Notes on character degrees quotients of finite groups. J Math 2002; 22(2): 217–220

[26] Qian G. Character degrees quotients and structures of finite groups. Chinese Ann Math Ser A 2005; 26(3): 307–312

[27] Qian G, Wang Y, Wei H. Co-degrees of irreducible characters in finite groups. J Algebra 2007; 312(2): 946–955

[28] Qian G. A note on element orders and character codegrees. Arch Math (Basel) 2011; 97(2): 99–103

[29] Qian G. A character theoretic criterion for p-closed group. Israel J Math 2012; 190: 401–412

[30] Qian G. A note on p-parts of character degrees. Bull Lond Math Soc 2018; 50(4): 663–666

[31] Qian G. Element orders and character codegrees. Bull Lond Math Soc 2021; 53(3): 820–824

[32] Qian G, Yang Y. Nonsolvable groups with no prime dividing three character degrees. J Algebra 2015; 436: 145–160

[33] Qian G, Zhou Y. On p-parts of character codegrees. J Group Theory 2021; 24(5): 1005–1018

[34] Ren Y. On character degrees quotients of finite groups. Acta Math Sinica Chin Ser 1995; 38(2): 164–170

[35] Sayanjali Z, Akhlaghi Z, Khosravi B. On the codegrees of finite groups. Comm Algebra 2020; 48(3): 1327–1332

[36] Tong-Viet H P. Brauer characters and normal Sylow p-subgroups. J Algebra 2018; 503: 265–276

[37] Xiong H. Finite groups whose character graphs associated with codegrees have no triangles. Algebra Colloq 2016; 23(1): 15–22

[38] Yang D, Gao C, Lv H. Finite groups with special codegrees. Arch Math (Basel) 2020; 115(2): 121–130

[39] Yang Y. On analogues of Huppert’s conjecture. Bull Aust Math Soc 2021; 104(2): 272–277

[40] Yang Y, Qian G. The analog of Huppert’s conjecture on character codegrees. J Algebra 2017; 478: 215–219

[41] Yang Y, Qian G. On p-parts of character degrees and conjugacy class sizes of finite groups. Adv Math 2018; 328: 356–366

[42] Zhang J. Arithmetical conditions on element orders and group structure. Proc Amer Math Soc 1995; 123(1): 39–44

RIGHTS & PERMISSIONS

Higher Education Press 2023