Embedding 2-groups in groups generated by involutions
Embedding 2-groups in groups generated by involutions
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将 2 组嵌入由对合生成的组中
DOI：
10.1016/j.jalgebra.2006.01.018
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发表时间：
2006
期刊：
影响因子：
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通讯作者：
N. Boston
中科院分区：
文献类型：
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作者：
N. Boston
本文讨论了一个群论问题，产生于数论。也就是说，给定一个二次域K和一个非分歧扩张L/K，它是Q上的Galois，Gal（L/K）的结构是否决定了Gal（L/Q）的结构？由于Q没有非平凡的非分歧扩张，Gal（L/Q）由它的惯性子群生成。由于L/K是非分歧的，所以这些惯性子群的阶为1或2.用群论的术语，给定一个有限或profinite群H，我们想把那些包含H作为指数为2的子群并且由G中H之外的对合生成的群G分类.我的论文与Leedham-Green [3]表明，与Lemmermeyer的一个猜想相反，存在无限族的有限2-群H，对于它不存在这样的G。在这里，我们将这个结果推广到表明，对于2-生成的H，如果这样的G存在，那么它是唯一的。
This paper addresses a group theory problem that arises from number theory. Namely, given a quadratic field K and an unramified extension L/K which is Galois over Q, does the structure of Gal (L/K) determine that of Gal (L/Q)? Since Q has no nontrivial unramified extensions, Gal (L/Q) is generated by its inertia subgroups. Since L/K is unramified, these inertia subgroups have order 1 or 2.In group-theoretical terms, given a finite or profinite group H, we want to classify those groups G that contain H as a subgroup of index 2 and that are generated by the involutions in G outside H. My paper with Leedham-Green [3] showed that, contrary to a conjecture of Lemmermeyer, there exist infinite families of finite 2-groups H for which no such G exists. Here we extend this result to show that for 2-generated H, if such a G exists, then it is unique.