## 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:

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:

## Introduction

Huppert’s – 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 Malle^{1}. 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 j^{2}. For comprehensive background information, consider works by Thomas M. Keller^{3}, Yong Yang^{4}, and Suzanne M. Seager^{5}. Notably, this dual problem was considered for character codegress:

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

Then the set of codegrees can be indicated by

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. Wolf^{3},

One of the principal proofs utilized to justify this proposition was the inequality bounding the derived length of a finite group G^{3}. 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

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 Actions*^{4}, 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

**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. Keller^{4}:

**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 and such that

This theorem holds true, but *Orbits in Finite Group Actions*^{4} 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 , and as , we see that there exist universal constants A and B such that

Now the Corollary in Section 6

^{6}yields that either

– in which case we are obviously done – or

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 and remains unknown. Optimizing values of and 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:

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 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*

Since , then

We first suppose Then

Finally, we may reduce *Inequality(1)* into

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)**

Substitute from **Theorem I** c = 36.44

Same logic as before. First suppose then again

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

Don’t forget to check the 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 Moretó^{2}, we obtain

**Theorem B.**

**Theorem B** *Proof.*

For the sake of completion, we replicate a more concise proof with essentially the same method used by Alexander Moretó^{2}, but of course with the more optimal constants we found. To continue, first consider **Theorem II** again proposed previously by Alexander Moretó^{2}:

**Theorem II**. Let be a prime. If is a finite solvable group with then

**Theorem II** *Proof*. If , we first define

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 . According to Gschutz’s theorem [10, Theorem 1.12]^{3}, splits over . Therefore, we can decompose as the semidirect product , where is a completely reducible faithful -module over the field with elements. We write as a direct sum of irreducible -modules . It is important to note that and acts faithfully and irreducibly on for every . Furthermore, embeds into , so

For each , write , where is the direct sum of the remaining ‘s.

Notice that . Thus, the semi-direct product of acting on is isomorphic to a quotient of . Specifically, . Also, for every , divides . Thus, divides for all . Additionally, since is the unique minimal normal subgroup of , all the characters in are faithful, implying that . Furthermore, based on [8, Theorem 6.18]^{1}, for all , extends to . From Clifford’s correspondence, it follows from [8, Theorem 6.16]^{1} that

Therefore, we may conclude that

Moreover, since acts faithfully and irreducibly on , it also acts faithfully and irreducibly on the dual group as stated in [Proposition 12.1]^{1}. Thus, by **Theorem A**,

Therefore, we also have

As noted by A. Moretó^{7}, it is unclear whether the derived length of a -group can be bounded by the number of character codegrees, making it difficult to remove in the statement of **Theorem II**. Next, we proceed to proving **Theorem B**, which requires the following lemma:

**Lemma 1-1.** Let be a group. Suppose that for every prime , has at maximum character codegrees divisible by . Let denote the normal subgroups of such that is nilpotent. Then we have

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

Here, is a Sylow -subgroup of for each and is a Hall -complement. Observe that the codegree of a linear character of order is exactly for each . Now, take of order for each . Note that for every , which implies . For , we adopt the convention that , so the statement remains valid.

Now let be characters lying over for each . Notice that has codegree a multiple of by **Lemma 1-1**. Also observe that if then does not divide , so the codegrees of are pairwise different. Therefore, we have found different character codegrees that are multiples of . Thus , as wanted.

Finally, let be a prime divisor of . We define as the inverse image in of . Applying **Theorem II** to we conclude that

By basic group theory, we know that the intersection of over all prime divisors of is contained in the Fitting subgroup . Therefore

Thus we deduce that there exist normal subgroups of , , such that , and for each the quotient 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 is at maximum . Therefore, when we let k = z, the results follow:

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 – Conjecture.

## References

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