On a chain of Butler groups

• Main Authors: Chang, Yuan-Rung, 張員榮
• Other Authors: Lee, Wu-Yen
• Format: Others
• Language: zh-TW
• Published: 1998
• Online Access: http://ndltd.ncl.edu.tw/handle/72403392469880494733

博士 === 淡江大學 === 數學學系 === 86 === 1937年德國數學家貝爾(Baer)對有限秩完全分解群找出了一個完全不變量 集。 自此以後, 有限秩非撓群的結構 理論反倒停滯不前了, 這其中瓶頸在於無法找到一個適當的中介族群, 這一族群要足夠大到能涵蓋一些有趣的例子, 而且要足夠小到我們寄望能瞭解其構造, 從中能夠加以分類, 並希望能發展出一套技巧以提供一些合理的問題。\par 1965年英國數學家巴特勒(Butler)提出了這個族群, 即有限秩完全分解群 的純子群; 巴特勒證明了完全分解群的純子 群的充要條件是此群乃為有限秩完全分解群的同態像。 同時他也藉這類群的構造之型(type)和其中的一些不變量子群來定位此類 群。\par 這一族群在有限秩非撓交換群的 理論中, 從此扮演了舉足輕重的角色。 這也正是我們數學界所要找的族群, 此類群後來被賴迪(Lady)稱為巴特勒 群(Butler group)。\par 過去這一、二十年, 有些人在探討兩類比較特殊的巴特勒群。設 $A=( A_1,\ldots$, $A_n)$ 是一組 $n$-分量 $Q$ 的子群, $G(A)=\{(a_1,\ldots,a_n)|a_1+a_2+ \cdots +a_n=0\}$ 及 $G\langle A\rangle =A_1 \oplus \cdots\oplus A_n/\langle (a,\ldots, a)\rangle_*$, $a\in \cap_{1\le i\le n} A_i$。1984年, 瑞屈曼(Richman)在文獻[14]中, 詳述 $G(A)$ 類群的性質, 我們也將在本論文中, 對 $G\langle A\rangle$ 的性質做一表述。\par 巴特勒群族包括了所有完全分解群。若對相異數對 $\{i,j\}$, $A_i+ A_j$ 的型為不可比的話, 那麼 $T_A=\{ $type $(A_i+\cap_{j\not=i} A_j)|i=1\cdots n\}$ 形成了 $G\langle A\rangle$ 的準同構不變量的 完全集。 我們在本篇論文的第二章裡將更 進一步對強不可分解群發展出同構不變量的完全集。\par 1974年, 賴迪在文獻[12]中介紹了調節子群(regulating subgroup)的概 念。 而在1981年, 大衛阿諾德(David Arnold)在文 獻[1]中, 建立了 $B_0$-群與 $B_1$-群皆為巴特勒群的概念。其中發現了: 若一個巴特勒群是 $B_0$- 群, 則它是自身唯一的調節子群, 且 若一個巴特勒群是 $B_1$-群, 則其每個調節子群均為 $B_0$-群。由此可見, 調節子群的概念對於測試巴特勒群的結構而言, 扮 演了重要角色。 本篇論文的第三章, 我們將以調節子群為基礎, 發展出巴特勒群子族類的一個嚴格遞增鏈。 \par \noindent {\it Abstract:} \bigskip\vskip 0.3 cm R. Baer, in 1937, gave a complete set of invariants for finite rank completely decomposable groups. After such a promising start, the theory of the structure of finite rank torsion free groups has become stagnant. One of the difficulties has been the absence of a suitable intermediate class of groups, i.e., a class large enough to contain interesting examples, small enough that there is some hope of understanding the structure of groups in the class, and admitting enough different characterizations to provide a variety of techniques and reasonable problems.\smallskip\par The class of pure subgroups of finite rank completely decomposable groups plays an important role in the theory of finite rank torsion free abelian groups. This class was first presented by M.C.R. Butler ([9]) in 1965 and then named after him in Lady [13]. \smallskip\par Butler proved that a torsion free group is a pure subgroup of a finite rank completely decomposable group if and only if it is the homomorphic image of a finite rank completely decomposable group. He also gave a characterization of these groups in term of types and their associated fully invariant subgroups.\smallskip\par For an n-tuple $A = (A_{1},\cdots,A_{n})$ of subgroups of {\bf Q}, we are concerned with the group $G \langle A \rangle = (A_{1} \oplus \cdots \oplus A_{n})/{\langle (a,\cdots,a) \rangle}_{*}$, $a \in {{\bigcap}_{1 \le i \le n}}{A_{i}}$. The group $G \langle A \rangle$ is the dual of the rank-(n-1) Butler group $G(A) = \{(a_{1},\cdots,a_{n}) \in {A_{1} \oplus \cdots \oplus A_{n}} \mid {{\sum}_{i=1}^{n}}{a_{i}} = 0\}$. In [14], Richman described some properties of $G(A)$. In this paper, we will provide some properties of $G\langle A\rangle$.\smallskip\par This class of Butler groups contains all completely decomposable groups. If the types of $A_{i} + A_{j}$ are incomparable for distinct pairs $\{i,j\}$, then $T_{A} = \{$type$(A_{i} + {\bigcap_{j \ne i}}{A_{j}}) \mid i =1,\cdots,n\}$ forms a complete set of quasi-isomorphism invariants for $G\langle A\rangle$. Furthermore, we develop a complete set of isomorphism invariants for those strongly indecomposable groups on chapter 2 in this paper.\smallskip\par In [12], Lady introduced the concept of a regulating subgroup of an almost completely decomposable group which turned out to be a completely decomposable subgroup of minimal index. The notation of regulating subgroups is generalized to the class of Butler groups.\smallskip\par David M. Arnold, in 1981, gave us the concepts of $B_{0}$-group and $B_{1}$-group which are Butler groups. If a Butler group is a $B_{0}$-group, then it is a unique regulating subgroup of itself. If a Butler group is a $B_{1}$-group then each regulating subgroup of it is a $B_{0}$-group. Hence, the concept of a regulating subgroups plays an important role to examine the structures of Butler groups. On chapter 3 in this paper, we will develop a strictly increasing chain of subclasses of Butler groups in terms of regulating subgroups. \smallskip\par