Definitions of a group and a field by independent postulates
Definitions of a group and a field by independent postulates
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通过独立假设定义群和域
DOI：
10.1090/s0002-9947-1905-1500706-2
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发表时间：
1905
影响因子：
1.3
通讯作者：
L. Dickson
中科院分区：
文献类型：
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作者：
L. Dickson
1. 这里给出的一般抽象群的简单定义与摩尔教授的两个定义（Transactions，vol.1）的起源和特征有关。 3 (1902)，第 485-492 页。在添加内容出现前几天，《交易》，卷。 5（1904），第。 549，摩尔教授在他的论文中对我说，他的一个与逆相关的假设是多余的，这意味着他的第二个定义的假设 ( 3" )。我认为参考的是他的第一个定义，并试图重建冗余的证明，那里不存在。这种尝试导致我改变他的假设 (4;) 以读取 aa'r = ir 而不是 a',a = ir 并注意到假设 (3;) 在更改后的集合中变得冗余，从而获得随后我了解到，摩尔教授在证明他的第二个定义中的（3”）冗余时，已经获得了足以建立当前定义的关系J，但没有应用它们来建立定义本身。当前的一般群假设具有理想的性质，即它们在特殊群的假设集中保持独立，其特殊性要么是在元素数量（n>1）的方向上，要么是它们的交换性（§§3-5）。字段的定义（§ 6）基于当前组的定义，比早期的定义具有明显的优势。 § 在集合是有限的，或形成可枚举无穷大，或不可枚举无穷大的假设下，域的假设保持独立。
1. The simple definition here given for a general abstract group relates as to origin and character to Professor Moore's two definitions, Transactions, vol. 3 (1902), pp. 485-492. A few days before the appearance of the addition, Transactions, vol. 5 (1904),p. 549, to his paper, Professor Moore remarked to me that one of his postulates relating to an inverse was redundant, meaning postulate ( 3" ) of his second definition. I thought the reference was to his first definition and attempted to reconstruct the proof of a redundancy, there absent. This attempt led me to alter his postulate (4;) to read aa'r = ir instead of a',a = ir and to note that postulate (3;) becomes redundant in the altered set, thus obtaining the present definition. Subsequently I learned that Professor Moore, in his proof of the redundancy of (3") in his second definition, had obtained relations J sufficient to establish the present definition but had not applied them to set up the definition itself. The present postulates for a general group possess the desirable property that they remain independent within sets of postulates for special classes of groups, the specialization being either in the direction of the number ( n > 1 ) of elements or their commutativity (§§ 3-5). The definition of a field (§ 6), based on the present definition of a group, has evident advantages over the earlier definitions. § The postulates for a field remain independent under an assumption that the set is finite, or forms an enumerable infinitude, or a non-enumerable infinitude.