Finite groups with exactly threep-regular classes
Finite groups with exactly threep-regular classes
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DOI：
10.1007/bf01189995
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发表时间：
1991-08
影响因子：
0.6
通讯作者：
Y. Ninomiya
中科院分区：
文献类型：
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作者：
Y. Ninomiya
1.引言。在本文中，G始终表示有限群。设p是素数，记为Rp，(G)G中p-正则类的个数。如果k是G的特征p的分裂域，则众所周知，Rp，(G)等于非同构的单kg-模的个数。在G是p-可解的情况下，Kg的主块Bo与群代数Kg/Of(G)同构，因此非同构单Bo-模的个数等于Rf(G/Of(G))。在这一点上，我们在[6]和[7]中确定了恰好有两个或恰好三个p-正则类的p-可解群的结构。特别地，这些结果暗示了当Rf(G)=2或3的p-可解群G是可解的。但如果Rf(G)=2，则G总是可解的，的确，如果Rf(G)=2，那么，当G只有一个非平凡的p-正则类时，G的阶的f-部分是不同于p的素数的幂。3])，G是可解的。因此，在没有Gp-可解的假设下，文[6]中的定理A成立。相反，具有Rp，(G)=3的群G并不总是可解的。事实上，存在恰好有三个p-正则类的不可解群。本文的目的是给出恰好有三个p-正则类的有限不可解群的结构。我们定理的证明依赖于有限单群的分类定理。
1. Introduction. Throughout this paper G will always denote a finite group. Let p be a prime number and denote by rp,(G) the number of p-regular classes in G. If k is a splitting field for G of characteristic p then, as is well-known, rp,(G) is equal to the number of non-isomorphic simple kG-modules. In the case when G is p-solvable, the principal block B o of kG is isomorphic to the group algebra kG/Of (G), and so the number of non-isomorphic simple Bo-modules is equal to rf (G/Of (G)). In this point of view, we determined in [6] and [7] the structure of p-solvable groups which have exactly two or exactly three p-regular classes. In particular, these results imply that a p-solvable group G with rf (G)= 2 or 3 is solvable. But if rf (G)= 2 then G is always solvable, Indeed, if rf (G)= 2 then, as G has only one nontrivial p-regular class, the f-part of the order of G is a power of a prime distinct from p. Hence, by Burnside p" qb-theorem ([2, Theorem 4.3. 3]), G is solvable. Therefore Theorem A in [6] holds without the assumption of G p-solvable. On the contrary, a group G with rp,(G)= 3 is not always solvable. In fact, there exist nonsolvable groups with exactly three p-regular classes. The purpose of this note is to give the structure of finite nonsolvable groups with exactly three p-regular classes. The proof of our theorem depends on the classification theorem of finite simple groups.