A Dual Version Of Huppert’s Conjecture: An Improved Bound Of Character Codegrees From Derived Length

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Abstract

Define G as a finite group and p as any prime. Now if z is a positive integer such that the number of character codegrees of G that are divisible by p is maximum at z, then the number of prime divisors of |G| is restricted by a function of z. This conjecture was proved for solvable groups and as a consequence, the set of character codegress in the dual problem acquired a nearly quadratic bound:

    \[|\text{cod}(G)| \le 24z^2\log_2{(z)} + 390z^2+1\]


This paper sought to optimize this restriction for character codegrees by obtaining a better constraint for derived length. Specifically, by employing first a general proof and then a proof by exhaustion, the results of this paper reduces the bound within:

    \[|\text{cod}(G)| \le 2z^2\log_2{(\log_2{(2z)})}+13.5z^2+1\]

Introduction

Huppert’s \rho\sigma Conjecture is one of the major open problems on character degrees of finite groups. There are many investigations on character codegress, in which some ruminated about a dual problem. For instance, let G be a finite group and denote the set of degrees of all complex irreducible characters of G as cd(G). Now if an integer j such that that for every prime p, at most j members cd(G) is divisible by p, then there exist a function of j that bounds |cd(G)|, as presented by Alexander Moretó and Gunter Malle1. In Particularly, in more recent years, Alexander Moretó established the proof that if G is solvable, then |cd(G)| is constrained by a quadratic function of j2. For comprehensive background information, consider works by Thomas M. Keller3, Yong Yang4, and Suzanne M. Seager5. Notably, this dual problem was considered for character codegress:

Let Irr(G) denote the set of all complex irreducible characters of G. Suppose \chi \in Irr(G), then the codegrees of \chi is rendered by

    \[|\text{cod}(\chi)| = \frac{\text{|G:Ker$\chi$|}}{\text{$\chi$(1)}}\]

Then the set of codegrees can be indicated by

    \[\text{cod}(G) = {\text{cod}(\chi) ~ \textbar{~} \chi \in Irr(G)}\]

Many researchers have inquired about this, and as of the most recent progress (see \textit{Representation Of Solvable Groups}3 for details), the constraint of the dual problem for codegrees was reduced to nearly a quadratic bound:

Let z be a positive integer. Now consider a finite solvable group G with the property that for any prime p, at most z members of cod(G) can be divided by p. Then from Olaf Manz and Thomas R. Wolf3,

    \[|\text{cod}(G)| \le 24z^2\log_2{(z)} + 390z^2+1\]

One of the principal proofs utilized to justify this proposition was the inequality bounding the derived length of a finite group G3. Thus in order to reduce this bound of codegrees further, consider minimizing this constraint on derived length first:

Let G be a solvable group and V be a vector space. Suppose G acts irreducibly and faithfully on V over a finite field, then

    \[\text{dl}(G) \le 24\log_2{(m\text{*}(G, V))} + 389\]

Here dl(G) denotes the derived length of G, that is the length of the derived series of the group. Note that the length of the series refers to the number of successive inclusions, thus the length is one less than the actual number of subgroups in the derived series. m*(G, V) indicates the number of orbit sizes of G on V that exceed 1. Recognize that the orbit sizes and the number of orbits are of the utmost importance in lowering this constraint. Along with a keen examination in Orbits in Finite Group Actions4, the derived length of G may easily be lowered, ultimately leading to a more optimal bound of character codegrees.

Theorems

Theorem A

Let G be a solvable group and m denote the number of nontrivial orbits, then

    \[\textnormal{dl} (G) \le 12.37 + 2\log_{2}(\log_{2}(m))\]

Theorem A: Proof

To acquire Theorem A, first consider the relationship between the derived length of a solvable group with the dimension and cardinality of a finite faithful irreducible G-module. These relationship will appear to be helpful in establishing Theorem A, as various substitutions can be made from these established inequalities. Thus to continue further, first consider Theorem I: proposed by Thomas M. Keller4:

Theorem I. Let G be a solvable group and V a be a finite faithful irreducible G-module.
The number of orbits of G on V will be denoted as r(G,V). Then there exist universal constants C_1 and C_2 such that

    \[\textnormal{dl} (G) \le C_1 + C_2\log_{2}(\log_{2}(r(G,V)))\]

This theorem holds true, but Orbits in Finite Group Actions4 provides an incorrect proof. Here, a correct and concise proof will be rendered before advancing.

Theorem I: Proof.

From Theorem 3.12 (b)3 we have \text{dl}(G) \le 2\log_2{(2\cdot \text{dim}(V))}, and as \text{dim}(V) \le\log_2|V|, we see that there exist universal constants A and B such that

    \[\textnormal{dl} (G) \le A + B\log_{2}(\log_{2}|V|).\]

Now the Corollary in Section 66 yields that either

    \[\text{dl}(G) \le 7.22 + \frac 52 \log_3\left( \log_3\left(\frac{r(G,V)+1.43} {24^{\frac 13}}\right)^c~\right)\]

– in which case we are obviously done – or

    \[|V|\le\left(\frac{r(G,V)+1.43} {24^{\frac 13}}\right)^c\]

where c = 36.44 in both scenarios; but in the other case using also implies the assertion, and so the theorem is proved.}

Despite of proving the theorem correctly, the 2 constants C_1 and C_2 remains unknown. Optimizing values of C_1 and C_2 to their minimal values can lower the constraint of the proposed inequality, ultimately contributing to significant improvements in lowering the bound of the derived length. To achieve this, various substitutions will be made. However, before advancing further, first consider these 2 inequalities in the previous proof:

    \[{(1)}\text{dl}(G) \le 2\log_2{(2\cdot \text{dim}(V))}\]

Now compare these 2 correct inequalities with the equation of derived length in the introduction. Notice how there exists extreme similarities that may perhaps allow substitutions. The only issue is that the inequality of the derived length in the introduction involves the number of orbits, whereas for the 2 inequalities above, the number of orbit sizes are used to bound the derived length of the group G. Nevertheless, this is quite convenient for to fix since m*(G, V) can be directly substituted with r(G, V)-1 as r(G, V)-1 denotes the number of orbit sizes of G on V that exceeds 1 and m is the number of nontrivial orbits. Since there are 2 proposed inequalities, now we check each of them in order to obtain an optimal constraint:

Inequality – 1

    \[\text{dl}(G) &\le 7.22 + \frac 52 \log_3\left( \log_3\left(\frac{r(G,V)+1.43}{24^{\frac 13}}\right)\right)\]


Since m = r(G,V) - 1, then

    \[\textnormal{dl}(G) &\le 7.22+\frac 52\log_3{\left(\log_3{\left(\frac{m+2.43}{24^{\frac 13}}\right)}\right)}\]

    \[\le 7.22+\frac 52\log_3{\left(\log_3{\left(24^{-\frac{1}{3}} {(m+2.43)}\right)}\right)}\]

    \[\le 7.22 + \frac {5}{2}\log_3{\left( \frac{\log_2{\left(24^{-\frac{1}{3}} {(m+2.43)}\right)}}{\log_2{3}} \right)}\]

    \[\le 7.22+\frac {5}{2\log_2{3}} \log_2{\left( \frac{\log_2{\left(24^{-\frac{1}{3}} {(m+2.43)}\right)}}{\log_2{3}} \right)}\]

    \[\le 7.22+ 1.58 \log_2{\left( \frac{\log_2{\left(24^{-\frac{1}{3}} {(m+2.43)}\right)}}{\log_2{3}} \right)}\]

We first suppose m\ge 2. Then

    \[m\le 24^{\frac 13}m^{(\log_2{3})}-2.43\]

Finally, we may reduce Inequality(1) into

    \[\textnormal{dl}(G) \le 7.22+1.58\log_2{(\log_2{m})}\]

m=1 still needs to be checked, and it does fall within the asserted result. As for \textit{Inequality (1)}, a better result was rendered. However, \textit{Inequality (2)} still needs to fall under this constraint.\

Inequality (2)

    \[\text{dl}(G) &\le 2\log_2{(2\cdot \text{dim}(V))}\]

Substitute from Theorem I c = 36.44

    \[\textnormal{dl}(G) &\le 2\log_2{\left(2\log_2{\left(\frac{m+2.43}{24^{\frac 13}}\right)}^c~\right)}\]

    \[\le 2\log_2{\left((2c)\log_2{\left(\frac{m+2.43}{24^{\frac 13}}\right)}\right)}\]

    \[\le 2\log_2{(2c)}+2\log_2{\left(\log_2{\left(\frac{m+2.43}{24^{\frac 13}}\right)}\right)}\]

    \[\le 12.37+2\log_2{\left(\log_2{\left(\frac{m+2.43}{24^{\frac 13}}\right)}\right)}\]

Same logic as before. First suppose m\ge 2, then again m\le 24^{\frac 13}m-2.43,

which after simplification, the final result for Inequality (2) establishes Theorem A:

    \[\textnormal{dl} (G) \le 12.37 + 2\log_{2}(\log_{2}(m))\]

Don’t forget to check the m=1 case, which once more resides below the constraint, hence the theorem remains valid.

Although it may appear disappointing that a less optimal bound was obtained at last, the 2 unknown constants were nevertheless found. Now with this newly acquired constraint of derived length, the bound of codegrees has significantly been reduced, as repeating the proof by Alexander Moretó2, we obtain

Theorem B. |\text{cod}(G)| \le 2z^2\log_2{(\log_2{(2z)})}+13.5z^2+1

Theorem B Proof.

For the sake of completion, we replicate a more concise proof with essentially the same method used by Alexander Moretó2, but of course with the more optimal constants we found. To continue, first consider Theorem II again proposed previously by Alexander Moretó2:

Theorem II. Let p be a prime. If G is a finite solvable group with O_{p'}(G) = 1 then

    \[dl(G/O_{p}(G)) \leq 12.37 + \log_{2}\log_{2}|\text{cod}_{p}(G)|\]

Theorem II Proof. If N \in G, we first define

    \[Irr(G|N) = { \chi \in Irr(G) | \chi > \text{Ker } \chi }\]

Additionally, we use cd(G|N) and cod(G|N) to represent the sets of degrees and codegrees, respectively, of the characters in Irr(G|N).

We can assume \Phi(G) = 1. According to G\"{a}schutz’s theorem [10, Theorem 1.12]3, G splits over F(G) = O_p(G). Therefore, we can decompose G as the semidirect product G = HV, where V = O_p(G) is a completely reducible faithful H-module over the field with p elements. We write V = V_1 \oplus \cdots \oplus V_t as a direct sum of irreducible H-modules V_i. It is important to note that H \cong G/O_p(G) and H/C_H(V_i) acts faithfully and irreducibly on V_i for every i = 1, \ldots, n. Furthermore, H embeds into H/C_H(V_1) \times \cdots \times H/C_H(V_t), so

    \[dl(G/O_p(G)) = dl(H) \leq \max{dl(H/C_H(V_1)), \ldots, dl(H/C_H(V_t))}.\]


For each i, write V = V_i \oplus W_i, where W_i is the direct sum of the remaining V_i‘s.

Notice that G/W_i \cong HVi. Thus, the semi-direct product \Gamma of H/CH(V_i) acting on V_i is isomorphic to a quotient of G. Specifically, \text{cod}(\Gamma) \subseteq \text{cod}(G). Also, for every 1_{V_i} \neq \lambda \in \text{Irr}(V_i), p divides \text{cod}(\lambda). Thus, p divides \text{cod}(\chi) for all \chi \in \text{Irr}(\Gamma|V_i). Additionally, since V_i is the unique minimal normal subgroup of \Gamma, all the characters in \text{Irr}(G|V_i) are faithful, implying that |\text{cd}(\Gamma|V_i)| = |\text{cod}(\Gamma|V_i)|. Furthermore, based on [8, Theorem 6.18]1, for all 1\neq\lambda \in Irr(V_i), \lambda extends to I\Gamma(\lambda). From Clifford’s correspondence, it follows from [8, Theorem 6.16]1 that

    \[m^*(H/CH(V_i), Irr(V_i)) \subseteq cd(\Gamma | V_i).\]

Therefore, we may conclude that

    \begin{align*} m^*(H/CH(V_i), \text{Irr}(V_i)) &\leq |\text{cod}(\Gamma|V_i)| \\ &\leq |\text{codp}(\Gamma)| \\ &\leq |\text{codp}(G)| \end{align*}

Moreover, since H/CH(V_i) acts faithfully and irreducibly on V_i, it also acts faithfully and irreducibly on the dual group Irr(V_i) as stated in [Proposition 12.1]1. Thus, by Theorem A,

    \begin{align*} dl(H/CH(V_i)) &\leq 12.37 + 2\log_2\log_{2}m^*(H/CH(V_i), Irr(V_i)) \\ &\leq 12.37 + 2\log_2 \log_{2}|cod_{p}(G)|  \end{align*}

Therefore, we also have

    \begin{align*} dl(G/Op(G)) &\leq \max\{dl(H/CH(V1)),~.~.~.~, dl(H/CH(Vt))\} \\ &\leq 12.37 + 2\log_{2}\log_{2}|\text{$cod_{p}$}(G)| \\ \end{align*}

As noted by A. Moretó7, it is unclear whether the derived length of a p-group can be bounded by the number of character codegrees, making it difficult to remove O_p(G) in the statement of Theorem II. Next, we proceed to proving Theorem B, which requires the following lemma:

Lemma 1-1. Let G be a group. Suppose that for every prime p, G has at maximum k character codegrees divisible by p. Let L \leq K denote the normal subgroups of G such that K/L is nilpotent. Then we have

    \[|\pi(K/L) - \pi(G/K)| \leq k\]

Lemma 1-1 Proof. We need to show that the number of “new” prime divisors appearing in K/L is at maximum k. Let us define

    \[\pi(K/L) - \pi(G/K) = \{p_1, \ldots, p_t\} = \pi\]

    \[\textnormal{where}\ K/L = P_1/L \times \cdots \times P_t/L \times H/L.\]

Here, P_i/L is a Sylow p_i-subgroup of K/L for each i and H/L is a Hall \pi-complement. Observe that the codegree of a linear character of order p_1 \cdots p_i is exactly p_1 \cdots p_i for each i. Now, take \lambda_i \in \text{Irr}(K/L) of order p_1 \cdots p_i for each i. Note that P_{i+1} \cdots P_tH \leq \text{Ker} \lambda_i for every i < t, which implies \lambda_i \in \text{Irr}(K/P_{i+1} \cdots PtH). For i = t, we adopt the convention that P_{i+1} \cdots P_tH = H, so the statement remains valid.

Now let \chi_i \in \text{Irr}(G) be characters lying over \lambda_i for each i. Notice that \chi_i \in \text{Irr}(G/P_{i+1} \cdot \ldots \cdot P_tH) has codegree a multiple of p_1 \ldots p_i by Lemma 1-1. Also observe that if 1 \leq j < k \leq t then p_k does not divide |G/P_{j+1} \cdot \ldots \cdot PtH|, so the codegrees of \chi_1, \ldots, \chi_t are pairwise different. Therefore, we have found t different character codegrees that are multiples of p_1. Thus t \leq k, as wanted.

Finally, let p be a prime divisor of |G|. We define O_{p'},<em>{p}(G) as the inverse image in G of O</em>{p}(G/O_{p'}(G)). Applying Theorem II to G/O_{p'}(G) we conclude that

    \begin{align*}     dl(G/Op',p(G)) &\leq 12.37 + 2 \log_{2}\log_{2}|\text{cod}p(G)| \\     & \leq 12.37 + 2 \log_{2}\log_{2}z  \end{align*}

By basic group theory, we know that the intersection of O_{p'},_{p}(G) over all prime divisors of |G| is contained in the Fitting subgroup F(G). Therefore

    \begin{align*} dl(G/F(G)) & \leq dl(G/ \bigcap_{p||G|} O_{p'},_{p}(G)) \\ &\leq \max_{p||G|} dl(G/O_{p'},_{p}(G)) \\ &\leq 12.37 + 2 \log_{2}\log_{2}z  \end{align*}

Thus we deduce that there exist normal subgroups of G, 1=N_{0} \leq N_{1} \leq \ldots \leq N_{l-1} \leq N_{l} = G, such that l \leq 12.37 + 2 \log_{2}\log_{2} k, and for each i the quotient N_{i+1}/N_{i} is nilpotent (and we may further assume that it is abelian for i > 0). Also, by the Lemma 1-1, the number of “new” prime divisors in every factor of N_{i+1}/N_{i} is at maximum k. Therefore, when we let k = z, the results follow:

    \[\textbf{Theorem B.}~~|\text{cod}(G)| \le 2z^2\log_2{(\log_2{(2z)})}+13.5z^2+1\]

This bound may perhaps be nowhere close to optimal. One contribution of this paper is the demonstration of a more complete picture on the Tightness of finite groups. When discussing group structures, the term Tightness refers to how restricted a group’s properties and operations are. Both Derived Length and character codegrees are important aspects that describe a group’s Tightness. The largest contribution of this paper is probably the demonstration of how crucial a role derived length plays in character codegrees as well as the significance of the specificity of the constants. Future improvements may be heavily relied on further reducing the 2 constants for the derived length, as this way, along with further investigations in orbit numbers and orbit sizes, more insights of character codegrees may be discovered, ultimately leading to a better comprehension of dual problems for Huppert’s \rho\sigma Conjecture.

References

  1. Moretó, G. Malle, A dual version of Huppert’s rho-sigma conjecture. International Mathematics Research Notices. 10, 1093 (2007). [] [] [] []
  2. A dual version of Huppert’s \rho\sigma conjecture for character codegrees. Forum Mathematicum, 34 – 2, 425 – 430 (2022). [] [] [] []
  3. O. Manz, T. R. Wolf, Representation of Solvable Groups. [] [] [] [] [] []
  4. T. M. Keller, Orbits in Finite Group Actions. (Email To Request Access). [] [] [] []
  5. T. M. Keller, Y. Yang Orbits of finite solvable groups on characters. Arxiv Math. 1208, 6024 (2012). []
  6. S. M. Seager A bound on the rank of primitive permutation groups. Journal Of Algebra. 116 – 2, 342 – 352 (1988). []
  7. A. Moretó Character degrees, character codegrees and nilpotence class of p-groups. Communications in Algebra. 50 – 2, 803 – 808 (2021). []

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