On linearly ordered groups
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1948 年 1 巻 1 号 p. 1-9
詳細
- Published: 1948 Received: 1947/09/01 Available on J-STAGE: 2006/08/29 Accepted: - Advance online publication: - Revised: -
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訂正情報
訂正日: 2006/08/29 訂正理由: - 訂正箇所: 引用文献情報 訂正内容: Wrong : 1) An element a is called positive in a 1. o. group, if a>e, where e is the unit of the group.
2) Everett, C. J. and Ulam, S. On ordered groups, Trans. Amer. Math. Soc. 57 (1945).
3) e α is the unit of. G α.
4) In the case of the restricted direct product we can define a linear order in another way. Cf. the first example at the end of the paper.
5) We can prove more generally (by making use of subsequent lemmas) that a linear order can be defined in such a group, which has no element of finite order and has a finite or transfinite ascending central chain.
6) Cf. H. Zassenhaus, Lehrbuch der Gruppentheorie (1937), p. 119 or E. Witt, Treue Darstellung Liescher Ringe, Crelle Jour., 177 (1937).
7) E. Witt, 1, c, 6).
8) The bracket means the commutator group.
9) E. Witt, 1, c. 6).
10) G. Birkhoff, Moore-Smith convergence in general topology, Ann. Math., 38 (1937).
11) If L is particularly a finite set, G is surely solvable in thee usual sense. Cf. Example 3, 6 below.
12) For the generalization of nilpotent groups cf R. Baer, Nilpotent groups and their generalizations, Trans. Math. Amer. Math. Soc., 47 (1940).
13) In this connection hold particularly remarkable relations between 1. o. groups and nilpotent groups, cf. 5).
14) Cf. Lemma 5.
Right : 1) An element a is called positive in a l. o. group, if a>e, where e is the unit of the group.
2) Everett, C. J. and Ulam, S. On ordered groups, Trans. Amer. Math. Soc. 57 (1945).
3) eα is the unit of. Gα.
4) In the case of the restricted direct product we can define a linear order in another way. Cf. the first example at the end of the paper.
5) We can prove more generally (by making use of subsequent lemmas) that a linear order can be defined in such a group, which has no element of finite order and has a finite or transfinite ascending central chain.
6) Cf. H. Zassenhaus, Lehrbuch der Gruppentheorie (1937), p. 119 or E. Witt, Treue Darstellung Liescher Ringe, Crelle Jour., 177 (1937).
7) E. Witt, l. c. 6).
8) The bracket means the commutator group.
9) E. Witt, l. c. 6).
10) G. Birkhoff, Moore-Smith convergence in general topology, Ann. Math., 38 (1937).
11) If L is particularly a finite set, G is surely solvable in the usual sense. Cf. Example 3, 6 below.
12) For the generalization of nilpotent groups cf R. Baer, Nilpotent groups and their generalizations, Trans. Math. Amer. Math. Soc., 47 (1940).
13) In this connection hold particularly remarkable relations between 1. o. groups and nilpotent groups, cf. 5).
14) Cf. Lemma 5.
訂正日: 2006/08/29 訂正理由: - 訂正箇所: PDFファイル 訂正内容: -
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