Counting characters above invariant characters in solvable groups

JOURNAL OF ALGEBRA

卷:472

Article

Cossey, James P.¹ 

[1] Univ Akron, Dept Theoret & Appl Math, Akron, OH 44325 USA

关键词: Brauer character;    Finite groups;    Representations;    Solvable groups;   

DOI  :  10.1016/j.jalgebra.2016.10.014

来源: Elsevier

【 摘 要 】

This paper discusses two related questions. First, given a G-invariant character 8 of a normal subgroup N of a solvable group, what can we say if the number of characters of above 8 is in some sense as small as possible? Isaacs and Navarro [5] have shown that under certain assumptions about primes dividing the order of the group, one can show that GIN must have a very particular structure. Here we show that these assumptions can be weakened to obtain results about all solvable groups. We also discuss a related question about blocks. For a prime p and a. p-block B of G, we let k(B) denote the number of ordinary characters in B. It is relatively easy to show that k(B) is bounded below by k(G, D), which is the number of conjugacy classes of G that intersect the defect group D of B. In this paper we ask what can be said if equality is achieved. We show that for p-solvable groups, if k(B) = k(G,D), then B is nilpotent and thus k(B) = vertical bar Irr(D)vertical bar. In addition, we show that this result holds for many blocks of arbitrary finite groups, including all blocks of the symmetric groups. We also extend a result on fully ramified coprime actions in [5]. (C) 2016 Elsevier Inc. All rights reserved.

【 授权许可】

Free