Finite 2-Groups with No Normal Elementary Abelian Subgroups of Order 8
Finite 2-Groups with No Normal Elementary Abelian Subgroups of Order 8
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DOI：
10.1006/jabr.2001.8972
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发表时间：
2001-12
影响因子：
0.9
通讯作者：
Z. Janko
中科院分区：
文献类型：
--
作者：
Z. Janko
第一个重要的结果是著名的4-生成元定理的麦克威廉姆斯[4]，其中断言任何子群这样的群体可以产生的四个元素。然而，这一结果并没有更多地说明这些群体的结构。在Konvisser [3]中，标题的群是在Frattini子群包含8阶初等交换子群的附加假设下确定的。这个决定是有点不幸的，因为所产生的群体是在发电机和关系，没有任何评论，并从这些很难披露的结构，这样的群体。但灾难还没有到来！Ustjuzaninov [5]在一篇很长的论文中断言，标题中的群G必须拥有正规亚循环子群N，使得G/N同构于8阶二面体群D8的子群。尽管这篇论文存在计算错误，但事实证明，这个结果毕竟是正确的！
The first important result about the groups of the title was the famous 4-generator theorem of MacWilliams [4], which asserts that any subgroup of such groups can be generated by four elements. However, this result does not say anything more about the structure of such groups. In Konvisser [3], the groups of the title are determined under the additional assumption that the Frattini subgroup contains an elementary abelian subgroup of order 8. This determination is somewhat unfortunate, because the resulting groups are given in terms of generators and relations without any comments and from these it is difficult to disclose the structure of such groups. But the catastrophe is yet to come!In a long paper, Ustjuzaninov [5] has asserted that the groups G of the title must possess a normal metacyclic subgroup N such that G/N is isomorphic to a subgroup of the dihedral group D8 of order 8. Even though this paper has computational errors, it turns out that this result is correct after all!