Finite p-groups with few minimal nonabelian subgroups
Finite p-groups with few minimal nonabelian subgroups
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DOI：
10.1016/j.jalgebra.2005.04.011
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发表时间：
2006-03
影响因子：
0.9
通讯作者：
Y. Berkovich
中科院分区：
文献类型：
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作者：
Y. Berkovich
1. 如果一个p群G是非abel的，但它的所有适当子群都是abel的，则称其为最小非abel群（为简洁起见，称为a1群）。推广这个概念，我们称p群G为an群，n∈n，如果G有一个指标pn−1的非abel子群，但它的指标pn的所有子群都是abel的。给定一个非abel p群G，有n∈n使得G是一个an群。例如，最大类和阶为pm的p群G是一个Am−2群；这个结果，由Blackburn的极大类p群理论（例如见[3，§9]）而来，并不是平凡的。（由此可知，如果pm阶的p群G中所有p3阶的子群都是阿贝尔的，则G的类最多为m−2。）L. Redei将a1群划分为素幂次（见引理1）。L. Kazarin在他未发表的论文中分类了素数幂阶的a2群（注意他的论述没有给出完整的证明；关于初等处理，参见[5，§§3-5]）。我认为本文是上述分类的一个重要应用
1. A p-group G is said to be minimal nonabelian (for brevity, A1-group), if G is nonabelian but all its proper subgroups are abelian. Generalizing this notion, we call a p-group G an An-group, n∈ N, if G possesses a nonabelian subgroup of index pn− 1 but all its subgroups of index pn are abelian. Given a nonabelian p-group G, there is n∈ N such that G is an An-group. For example, a p-group G of maximal class and order pm is an Am− 2-group; this result, following from Blackburn’s theory of p-groups of maximal class (see, for example,[3, § 9]), is not trivial.(It follows that if all subgroups of order p3 in a p-group G of order pm are abelian, then the class of G is at most m− 2.) A1-groups of prime power order were classified by L. Redei (see Lemma 1, below). A2-groups of prime power order were classified by L. Kazarin in his unpublished thesis (note that his exposition did not give a full proof; for an elementary treatment, see [5, §§ 3–5]). I consider this paper as a nontrivial application of the above classification and