Fundamentals of Group Theory: An Advanced Approach
Fundamentals of Group Theory: An Advanced Approach
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群论基础：高级方法
DOI：
10.1007/978-0-8176-8301-6
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发表时间：
2011
期刊：
影响因子：
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通讯作者：
Steven Roman
中科院分区：
文献类型：
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作者：
Steven Roman
群论基础提供了一个先进的看看基本理论的群体。该领域的标准主题与大量独特的内容一起涵盖。在讨论同构定理、商群和自由群时，强调了普遍性，并重点讨论了应用某些运算（如交集、提升和商）对“群扩展”的作用。某些概念，如次常态，群作用和链条件，可能比这个级别的其他文本早一点介绍，希望读者能更早适应这些概念。这项工作的一些其他功能包括：·伽罗瓦如何看待群体的历史。一个群的换位子群是否与该群的换位子集相同的问题，包括一个不相同的例子。·次正规联接属性，即两个次正规子群的联接是次正规的属性。·直接金额的取消。·Baer定理的一个完整证明，刻画了非交换群的性质，即它们的所有子群都是正规的。对p-群的结构进行了更深入的讨论，包括p-群中共轭的性质，证明了具有任何阶的唯一子群的p-群必须是循环的（p> 2）或循环的或广义四元数（p= 2），以及具有p^（n-1）阶元素的p^ n阶群的性质。用圈积讨论对称群的Sylow子群。·介绍用于刻画有限单群的技巧。·关于方程类和相对自由度的伯克霍夫定理。这本书是适合研究生班在群论，一部分研究生班在抽象代数或独立研究。它也可以由高级本科生阅读。这本书假设没有具体的背景，在群论，但确实假设一定程度的数学复杂性的一部分，读者。
Fundamentals of Group Theory provides an advanced look at the basic theory of groups. Standard topics in the field are covered alongside a great deal of unique content. There is an emphasis on universality when discussing the isomorphism theorems, quotient groups and free groups as well as a focus on the role of applying certain operations, such as intersection, lifting and quotient to a “group extension”. Certain concepts, such as subnormality, group actions and chain conditions are introduced perhaps a bit earlier than in other texts at this level, in the hopes that the reader would acclimate to these concepts earlier. Some additional features of the work include:· An historical look at how Galois viewed groups.· The problem of whether the commutator subgroup of a group is the same as the set of commutators of the group, including an example of when this is not the case.· The subnormal join property, that is, the property that the join of two subnormal subgroups is subnormal.· Cancellation in direct sums.· A complete proof of the theorem of Baer characterizing nonabelian groups with the property that all of their subgroups are normal.· A somewhat more in depth discussion of the structure of p-groups, including the nature of conjugates in a p-group, a proof that a p-group with a unique subgroup of any order must be either cyclic (for p> 2) or else cyclic or generalized quaternion (for p= 2) and the nature of groups of order p^ n that have elements of order p^(n-1).· A discussion of the Sylow subgroups of the symmetric group in terms of wreath products.· An introduction to the techniques used to characterize finite simple groups.· Birkhoff's theorem on equational classes and relative freeness. This book is suitable for a graduate class in group theory, part of a graduate class in abstract algebra or for independent study. It can also be read by advanced undergraduates. The book assumes no specific background in group theory, but does assume some level of mathematical sophistication on the part of the reader.