Abstract In this paper we first state a conjecture on the lower bound of the maximal height of characters in a p-block of a finite group. Then we show that our conjecture holds for all blocks of covering groups of a sporadic simple group, for all blocks of a quasi-simple group G with G / Z ( G ) isomorphic to A 6 , A 7 or a simple group of Lie type with an exceptional covering group, for all blocks of a symmetric group, as well as for all blocks of finite general linear or unitary groups. For the proof of the case with symmetric groups, it relates to an open question of Olsson on the existence of t-core partitions, where t ≥ 3 is an integer. As a byproduct, our investigation on heights of characters of the symmetric groups and of the general linear or unitary groups also gives evidence for the Isaacs-Moreto-Navarro-Tiep Conjecture, claiming that the number of distinct irreducible character degrees of a Sylow p-subgroup P of an arbitrary finite group G is at most one more than the number of irreducible character degrees of G that are multiples of p.

中文翻译：

---

关于有限群的字符高度

摘要 在本文中，我们首先对有限群的 p 块中字符的最大高度的下界进行了猜想。然后我们证明我们的猜想对于一个散发单群的所有覆盖群块，对于一个 G / Z ( G ) 同构于 A 6 , A 7 的拟单群 G 的所有块或一个 Lie 型单群都成立具有特殊覆盖群，对于对称群的所有块，以及有限一般线性或酉群的所有块。对于对称群情况的证明，它涉及到 Olsson 关于 t 核分区的存在性的一个悬而未决的问题，其中 t ≥ 3 是一个整数。作为副产品，我们对对称群和一般线性群或酉群的特征高度的研究也为 Isaacs-Moreto-Navarro-Tiep 猜想提供了证据，