Counting characters above invariant characters in solvable groups

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• 作者：James P. Cossey ^{cossey@uakron.edu}
• 关键词：20C20
• 刊名：Journal of Algebra
• 出版年：2017
• 出版时间：15 February 2017
• 年：2017
• 卷：472
• 期：Complete
• 页码：425-436
• 全文大小：332 K
• 卷排序：472

文摘

This paper discusses two related questions. First, given a G-invariant character θ of a normal subgroup N of a solvable group, what can we say if the number of characters of G above θ 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 G/NG/N 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)k(B) denote the number of ordinary characters in B  . It is relatively easy to show that k(B)k(B) is bounded below by k(G,D)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)k(B)=k(G,D), then B   is nilpotent and thus k(B)=|Irr(D)|k(B)=|Irr(D)|. 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].