A note on the number of irreducible characters in a $p$-block of a finite group
A note on the number of irreducible characters in a $p$-block of a finite group
复制标题
关于有限群的 $p$ 块中不可约字符数量的注释
DOI：
10.18910/6646
复制
发表时间：
1984
影响因子：
0.4
通讯作者：
M. Murai
中科院分区：
文献类型：
--
作者：
M. Murai
设G是一个有限群，n是| G |的除数，G的阶。设Ln(G)={x^G\x =l}。G. Frobenius给出了下面的猜想。(F)：如果Ln(G)\ =n>，则Ln(G)是G的正规子群。(1)为了证明(F)，我们可以假设n和\G\jn是相对素数。(2)设G为(F)的最小反例。那么G是一个单基。（见[1]，[12],[17]）由于有限单群的分类已经完成，可以通过(2)检验单群来验证(F)。事实上，一些作者（[1],[8],[9],[14],[16]）已经对某些类别的简单群进行了这样的验证。然而，从更普遍的角度来研究(F)也是可取的。本文的目的是提供这种调查的一个例子。对于群G和一撇p，我们用Gy表示，G的^'-元素的集合，或^-规则元素的集合，用G p表示p除以| G |的最高次幂。设\G\p'=\G\l\G\p。我们感兴趣的是下面(F)的特殊情况。
Let G be a finite group and n a divisor of | G |, the order of G. We set Ln(G)={x^G\x =l}. G. Frobenius gave the following conjecture. (F): If \Ln(G)\ =n> then Ln(G) is a normal subgroup of G. Concerning this problem the following results are known. (1) In order to prove (F) we may assume that n and \G\jn are relatively prime. (See [7], [12]) (2) Let G be a minimal counterexample to (F). Then G is a simple group. (See [1], [12], [17]) Since the classification of finite simple groups has been completed, it may be possible to verify (F) by checking simple groups, by using (2). Indeed such verifications have been carried out for certain classes of simple groups by several authors ([1], [8], [9], [14], [16]). However, it may also be desirable to investigate (F) from a more general standpoint. The purpose of this paper is to provide an example of such investigations. For a group G and a prime p, we denote by Gy, the set of ^'-elements, or ^-regular elements, of G and by \G\p the highest power of p dividing | G | . We set \G\p'=\G\l\G\p. We are interested in the following special case of (F).