The second homology group of a group; relations among commutators
The second homology group of a group; relations among commutators
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群的第二个同源群；
DOI：
10.1090/s0002-9939-1952-0049191-5
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发表时间：
1952
期刊：
影响因子：
--
通讯作者：
C. Miller
中科院分区：
文献类型：
--
作者：
C. Miller
我们关心的问题是分配一个群论解释的第二个同调群H2（G，J）的群G，与整数系数，J[1，p.486]。我们将定义一个新的群H（G），称为G的结合群，粗略地说，它是G中的扩张子所满足的所有关系的群，取模那些平凡或普遍满足的关系。(The提醒读者不要期望阿贝尔群的相关群必然为零;我们不认为“x和y可换意味着[x，y] = l”的陈述是一种关系。然后我们证明了H（G）^H2（G，J），使得H2（G，J）给出了G中的代数之间的关系在多大程度上不是泛关系的结果的度量。对于给定的群G，设（G，G）是所有对（x，y）上的自由群，其中x，y ∈ G。有一个自然同态{G，G）到[G，G]，它将（x，y）发送到[x，y}。如果wE（G，G），我们用[w]表示它在[G，G]中的像，并定义Z（G）为核，
We are concerned with the problem of assigning a group theoretic interpretation to the second homology group H2(G, J) of a group G, with integer coefficients, J[l, p. 486]. We shall define a new group, H(G), called the associated group of G, which is, roughly speaking, the group of all relations satisfied by commutators in G, taken modulo those relations which are trivially, or universally, satisfied. (The reader is cautioned not to expect that the associated group of an abelian group necessarily vanishes; we do not regard the statement "x and y commute implies [x, y] = l" as a relation.) We then show that H(G)^H2(G, J), so that H2(G, J) gives a measure of the extent to which relations among commutators in G fail to be consequences of universal relations. For a given group G, let (G, G) be the free group on all pairs (x, y), with x, y EG. There is a natural homomorphism of {G, G) onto [G, G] which sends (x, y) into [x, y}. If wE(G, G), we denote its image in [G, G] by [w], and define Z(G) to be the kernel,