Refinements for infinite direct decompositions of algebraic systems.
Refinements for infinite direct decompositions of algebraic systems.
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DOI：
10.2140/pjm.1964.14.797
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发表时间：
1964-09
影响因子：
0.6
通讯作者：
Peter Crawley;B. Jónsson
中科院分区：
文献类型：
--
作者：
Peter Crawley;B. Jónsson
介绍。具有一个主级数的算子群显然可以写成有限多个直接不可分解的可容许子群的直接积，经典的wedderburnremakkrull - schmidt定理断言这种表示在同构范围内是唯一的。这个定理的许多推广在文献中是已知的。因此，由Baer[1,2]的结果可知，如果算子群G的可容许中心满足极小和局部极大条件，则G的任意两个直接分解（具有任意多因子）具有同构细化。在不同的方向上，Crawley[4]表明，如果一个算子群G有一个直接分解，其中每个因子都有一个主级数，那么G的任意两个直接分解都有同构细化。对于算子群，得到了上述定理的一个一般推广：如果算子群G有一个直接分解，使得每个因子的可容许中心满足极小和局部极大条件，则G的任意两个直接分解都有中心同构改进。对于没有算子的群，我们得到了以下结果，消除了链条件的任何假设：如果群G（没有算子）有一个直接分解，使得每个因子的中心是可数的，并且每个因子中心的约简部分是一个有界阶的主成分的扭转群，那么G的任意两个直接分解都有中心同构的细化。
Introduction. An operator group with a principal series can obviously be written as a direct product of finitely many directly indecomposable admissible subgroups, and the classical WedderburnRemak-Krull-Schmidt Theorem asserts that this representation i& unique up to isomorphism. Numerous generalizations of this theorem are known in the literature. Thus it follows from results in Baer [1, 2] that if the admissible center of an operator group G satisfiesthe minimal and the local maximal conditions, then any two direct decompositions of G (with arbitrarily many factors) have isomorphic refinements. In a different direction, it is shown in Crawley [4] that if an operator group G has a direct decomposition each factor of which has a principal series, then any two direct decompositions of G haveisomorphic refinements. The results of this paper yield sufficient conditions for a group(with or without operators) to have the isomorphic refinement property* For operator groups a common generalization of the theorems mentioned above is obtained: If an operator group G has a direct decomposition such that the admissible center of each factor satisfies the minimal and local maximal conditions, then any two direct decompositions of G have centrally isomorphic refinements. For groups without operators we obtain the following result which eliminates any assumption of chain conditions: If a group G (without operators) has a direct decomposition such that the center of each factor is countable and the reduced part of the center of each factor is a torsion group with primary components of bounded order, then any two direct decompositions of G have centrally isomorphic refinements.