Ends of Groups and the Associated First Cohomology Groups
Ends of Groups and the Associated First Cohomology Groups
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群的端点和相关的第一上同调群
DOI：
10.1112/jlms/s2-6.1.81
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发表时间：
1972
影响因子：
1.2
通讯作者：
C. Houghton
中科院分区：
文献类型：
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作者：
C. Houghton
Hopf [7]和Freudenthal [6]定义了群的端点。我们将定义扩展到无限生成群，并推广了他们的一些结果。我们证明了一个非局部有限的群有1个、2个或无穷多个端点。这样的群有2个端点当且仅当它有一个有限指数的无限循环子群。一个群有1个端点，如果它有一个无穷指数的n-生成子群，其中包含一个在整个群中次正规且不是局部有限的子群f.由Specker [12]的结果可以得出n-生成群B的端点数与dimH 1（B，Z 2 B）有关. Swan [14]研究了任意环R的f ^（Bj-RB）。设A和B是群，A（B）是从B到A的有限支集函数f的B-群，其中fb（x）= f（x B ~ 1），对x，be B.根据Serre对群的非阿贝尔上同调的解释[9]，我们用与端有关的划分来描述H1（B，AW）。如果A是环R的加法群，则A（B）和RB之间存在B-同构，因此我们有Swan考虑的情形，即R生成的B。最后，我们利用我们的结果证明了：如果B有1个端点且表示为F/N，则F/Nm [N，N]的任何使N/Nm [N，N]按元素固定的自同构都是由N的一个元素诱导的内自同构.
The ends of a finitely generated group were defined by Hopf [7] and Freudenthal [6]. We extend the definition to infinitely generated groups and generalise some of their results. We show that a group which is not locally finite has 1, 2, or an infinite number of ends. Such a group has 2 ends if and only if it has an infinite cyclic subgroup of finite index. A group has 1 end if it has a finitely generated subgroup of infinite index containing a subgroup which is subnormal in the whole group and is not locally finite f.It follows from the results of Specker [12] that the number of ends of a finitely generated group B is related to dimH1 (B, Z2 B). Swan [14] has investigated/^(Bj-RB) for any ring R. Suppose that A and B are groups and let A {B) be the B-group of functions/from B to A with finite support, where fb (x)= f (xb~ 1) for x, beB. Following Serre's account of the non-Abelian cohomology of groups [9], we describe H1 (B, AW) in terms of partitions related to ends. If A is the additive group of the ring R, there is a B-isomorphism between A^ B) and RB and so we have the case considered by Swan for finitely generated B. Finally, we use our results to show that if B has 1 end and is presented as F/N then any automorphism of F/Nm [N, N] which leaves N/Nm [N, N] elementwise fixed is an inner automorphism induced by an element of N.