Finding minimal n-power extensions of groups.

机译：查找组的最小n次幂扩展。

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In this paper we consider the group-theoretic analogue of Galois Theory. That is, given a group G and a natural number n, we find groups H such that G can be embedded into H in such a way that every element of G has an nth root in H. It is not difficult to see (and we prove) that given any group G and natural number n such an extension of G exists. In this paper, we attempt to find the smallest order of such an extension, which we will call the minimal n-index of G. We answer the question completely for cyclic groups. Further, we examine certain ways of constructing new groups from old, in particular the direct and semidirect product, and determine how these constructions interact with the minimal n-index of G. We conclude with some conjectures regarding the minimal n-index and some questions to inspire further research.

机译：在本文中，我们考虑了伽罗瓦理论的群论类比。也就是说，给定一个组G和一个自然数n，我们找到组H，这样G可以嵌入到H中，使得G的每个元素都在H中有第n个根。证明）给定任何组G和自然数n都存在G的扩展。在本文中，我们试图找到这种扩展的最小阶，我们将其称为G的最小n索引。对于循环群，我们完全回答了这个问题。此外，我们研究了从旧的，特别是直接和半直接乘积构造新组的某些方法，并确定了这些构造如何与G的最小n指数相互作用。我们得出关于最小n指数的一些猜想和一些问题激发进一步的研究。

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作者
Taylor, William D.;
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作者单位

University of Nevada, Reno.;

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授予单位 University of Nevada, Reno.;
学科 Mathematics.
学位 M.S.
年度 2010
页码 40 p.
总页数 40
原文格式 PDF
正文语种 eng
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Finding minimal n-power extensions of groups.

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