On Analogues of Huppert’s Conjecture

Abstract

Let G be a finite group and $\chi $ be a character of G. The codegree of $\chi $ is $\text{codeg} (\chi ) ={|G: \ker \chi |}/{\chi (1)}$ . We write $\pi (G)$ for the set of prime divisors of $|G|$ , $\pi (\text{codeg} (\chi ))$ for the set of prime divisors of $\text{codeg} (\chi )$ and $\sigma (\text{codeg} (G))= \max \{|\pi (\text{codeg} (\chi ))| : \chi \in {\textrm {Irr}}(G)\}$ . We show that $|\pi (G)| \leq ({23}/{3}) \sigma (\text{codeg} (G))$ . This improves the main result of Yang and Qian [‘The analog of Huppert's conjecture on character codegrees’, J. Algebra478 (2017), 215–219].

Element orders and character codegrees

“…While both papers were published in the same journal and the same year, the techniques used in those two papers are quite different. This result has already been applied to study the codegree version of the Huppert's ‐ conjecture [13, 14]. The main goal of this paper is to substantially strengthen the main result of [11], which confirms a conjecture proposed in [11] for the case of solvable groups.…”

“…In [12], we have obtained many basic properties of codegree and also studied its related prime graph. Since then, this new definition of codegree has attracted quite amount of interest [1, 5, 8–11, 13, 14]. Among those, the most interesting finding perhaps is the relation between the character codegrees and the element orders [8, 11].…”

Two results on character codegrees

“…We note that the first question was answered in [15] and further improved in [16]. In this paper, we show that the second question is not always true in general and we obtain the best possible bound for |π(G)| under the condition σ c (G) = 2, 3, and 4 respectively.…”

New bounds for numbers of primes in element orders of finite groups

“…Yang [19] showed that 𝐾 ≤ 17∕3. A weaker (but simpler) proof that 𝐾 ≤ 7 was given by Hung and Yang [5].…”

“…He [9] showed that there is a function 𝐶(𝑛) such thatwhere 𝐶(𝑛) is bounded, and lim 𝑛→∞ 𝐶(𝑛) = 4. Of interest is determining the smallest constant 𝐾 such that 𝜌(𝑛) ≤ 𝐾𝑛, (𝑛 ≥ 1).Yang [19] showed that 𝐾 ≤ 17∕3. A weaker (but simpler) proof that 𝐾 ≤ 7 was given by Hung and Yang [5].…”