SOME REMARKS ON COMMUTATORS
SOME REMARKS ON COMMUTATORS
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关于换向器的一些说明
DOI：
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发表时间：
1951
期刊：
影响因子：
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通讯作者：
O. Ore
中科院分区：
文献类型：
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作者：
O. Ore
在一个群中，两个换位子的乘积不一定是换位子，因此一个给定群的换位子群不能定义为所有换位子的集合，而只能定义为由这些换位子生成的群。当换位子群的所有元素都是换位子时，关于这个问题的标准或研究似乎很少。下面证明了在有限对称群2n中，交错群的所有元素都是交换子;我们可以推广这一点，证明当n >5时，交错群中的所有元素都是An中元素的交换子。对于无限对称群的情况是不同的，因为我们得到：任何一个无限集对自身的一一对应是一个交换子。1.我们将开始作一些一般性的评论，这些评论既适用于有限的情况，也适用于无限的情况。任意集合S与其自身的任何一一对应T都可以唯一地写成对不相交的元素集合进行运算的圈的乘积
In a group the product of two commutators need not be a commutator, consequently the commutator group of a given group cannot be defined as the set of all commutators, but only as the group generated by these. There seems to exist very little in the way of criteria or investigations on the question when all elements of the commutator group are commutators. In the following it is shown that in the finite symmetric group 2n all elements of the alternating group are commutators; one can extend this and show that when n >5 all elements in the alternating group are commutators of elements in An. For the infinite symmetric group the situation is different since we obtain: Any one-to-one correspondence of an infinite set to itself is a commutator. 1. We shall begin by making a few general remarks which apply both to the finite and the infinite cases. Any one-to-one correspondence T of an arbitrary set S to itself can be written uniquely as a product of cycles which operate on disjoint sets of elements