Decomposition of a group with a single defining relation into a free product
Decomposition of a group with a single defining relation into a free product
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将具有单一定义关系的组分解为自由产品
DOI：
10.1090/s0002-9939-1955-0069174-1
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发表时间：
1955
影响因子：
2.2
通讯作者：
A. Shenitzer
中科院分区：
文献类型：
--
作者：
A. Shenitzer
设G是生成元为a，v = 1，*，n的群.自由群的任何自同构A在a上的应用，或者等价地，T-变换序列（定义如下）映射G在同构群G '上的应用。如果G由a的一组规定关系定义，则G'可以通过将原始关系转录为A-la来定义。即使G由一个单一的关系定义，也不知道具有单一定义关系并同构于给定关系的所有群的集合在多大程度上由变换A决定。然而，Grushko的定理[2 ]2意味着，至少G分解为它的两个真子群的自由积可以通过应用适当选择的A而变得明显。我们将证明，对于具有单一定义关系的G，J. H. C. Whitehead [1]给出了求A的构造性方法和G的自由不可分解性的简单判别法。定义和注释。(1)T变换通过对生成元a，，.，an的自由群F =F（al，*，an），我们表示以下形式的映射：
Let G be a group with generators a, v = 1, * , n. An application of any automorphism A of the free group on the a, or, equivalently, of a sequence of T-transformations (defined below) maps G upon an isomorphic group G'. If G is defined by a set of prescribed relations for the a, G' can be defined by transcribing the original relations in terms of the A-la,. Even if G is defined by a single relation, it is not known how far the set of all groups with a single defining relation and isomorphic to a given one is determined by the transformations A. However, Grushko's theorem [2 ]2 implies that at least the decomposibility of G into a free product of two of its proper subgroups can be made obvious by applying a properly chosen A. We shall show that for a G with a single defining relation a result of J. H. C. Whitehead [1] provides a constructive method for finding A and some simple tests for the free indecomposability of G. DEFINITIONS AND REMARKS. (1) T-transformations. By a T-transformation on the generators a,, ... , an of the free group F =F(al, * * *, an) we mean a mapping of the form: