Some applications of the theory of blocks of characters of finite groups IV
Some applications of the theory of blocks of characters of finite groups IV
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DOI：
10.1016/0021-8693(71)90006-8
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发表时间：
1964-12
影响因子：
0.9
通讯作者：
R. Brauer
中科院分区：
文献类型：
--
作者：
R. Brauer
在同题系列论文[3]中，给出了有限群特征标块理论的一些应用。这项工作将在这里和随后的一篇论文中继续。在第二节中，对该符号进行了解释。给出了一些假定结果，并证明了一些引理。第三节研究了块的核。这推广了文[31]中证明的关于主块的结果。同样，第四节讨论了[31]中定理3的扩展。我们关注的是某些类型的完全缺陷块。这类方法在研究具有拟二面体和有圈Sylow 2-子群的有限群中起着重要的作用。在[31]中已经证明，对于给定的素数p和给定的亏数d，任意有限群的亏数d的p-块落入有限个‘Yypes’中。这一调查以更精确的形式进行，并在第五节继续进行。从某种程度上说，这些结果为进一步研究区块提供了理由。在某些情况下，可以对可能的类型进行充分的讨论。这里我们处理第七节中p=2，d=2的情况，并利用这些结果得到G的阶公式。这个公式在[L]中是需要的。在其他情况下，可以对这些块进行全面讨论。这将在其他地方完成。前面的第六节包含了第七节中所需要的关于块的一些结果。我们还可以改进[411]关于具有阿贝尔缺陷组的块中的字符高度的结果。最后，第八节包含了一些关于逆块的相当简单的结果。这些在[i]中也是需要的。
In a series of papers of the same title [3], some applications have been given of the theory of blocks of characters of finite groups. This work will be continued here and in a subsequent paper. In Section II, the notation is explained. Some presupposed results are stated, and a number of lemmas are proved. In Section III, the kernel of a block is studied. This extends results proved in [31] for the principal block. Again, Section IV deals with extensions of Theorem 3 of [31]. We are concerned with certain types of blocks of full defect. Methods of this type play an important role in the investigation of finite groups with quasidihedral and wreathed Sylow 2-subgroup, cf.[I]. In particular, they will be needed in a continuation of [I], in order to develop the necessary character theory for the latter type of groups.It had already been shown in [31], that for a given prime p and a given defect d, the p-blocks of defect d of arbitrary finite groups fall into a finite number of ‘Yypes.” This investigation is covered in a more precise form and continued in Section V. In a way, the results provide a justification for a further study of blocks. There are cases in which a full discussion of the possible types can be given. We deal here with the case p= 2, d= 2 in Section VII and use the results to obtain a formula for the order of G. This formula is needed in [l]. There are other cases in which a full discussion of the blocks can be given. This will be done elsewhere. The preceding Section VI contains some results on blocks which are needed in Section VII. We can also improve results of [411] on the height of the characters in blocks with abelian defect groups. Finally, Section VIII contains some rather simple results on contragredient blocks. These too are needed in [I].