SHORT PROOFS OF SOME BASIC CHARACTERIZATION THEOREMS OF FINITE p-GROUP THEORY
SHORT PROOFS OF SOME BASIC CHARACTERIZATION THEOREMS OF FINITE p-GROUP THEORY
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DOI：
10.3336/gm.41.2.07
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发表时间：
2006-12
影响因子：
0.4
通讯作者：
Y. Berkovich
中科院分区：
文献类型：
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作者：
Y. Berkovich
对Blackburn、Huppert、Ito-Ohara、Janko、Taussky等人的基本定理进行了简短的证明。这些定理的所有证明都基于以下结果：如果G是一个非贝尔亚环p群，R是G0的一个固有G不变子群，则G=R不是亚环。第二部分运用了Blackburn的极大类p群理论。本文证明了p群G是极大类的当且仅当2 (G) = hx2gjo (x) = p2i是极大类。我们还证明了指数为>p的非循环p群G包含两个不同的>p阶的极大循环子群a和B，使得jA \ Bj = p，除非p = 2且G是二面体。
We oer short proofs of such basic results of nite p-group theory as theorems of Blackburn, Huppert, Ito-Ohara, Janko, Taussky. All proofs of those theorems are based on the following result: If G is a nonabelian metacyclic p-group and R is a proper G-invariant subgroup of G0, then G=R is not metacyclic. In the second part we use Blackburn's theory of p-groups of maximal class. Here we prove that a p-group G is of maximal class if and only if 2 (G) = hx 2 G j o(x) = p 2i is of maximal class. We also show that a noncyclic p-group G of exponent > p contains two distinct maximal cyclic subgroups A and B of orders > p such that jA \ Bj = p, unless p = 2 and G is dihedral.