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On Groups with Chain Conditions

Published online by Cambridge University Press:  20 November 2018

Bernhard Amberg
Bernhard Amberg*
Affiliation:
The University of Texas, Austin, Texas
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Our aim in this note is to generalize results of Baer in [3; 5]. In § 1 an arbitrary formation n is considered, the key result being Proposition 1.5. This is applied in § 2 to characterize various finiteness conditions, for example the classes of groups with maximum [minimum] condition on subgroups, subnormal subgroups, and normal subgroups respectively, or the class of (not necessarily soluble) polyminimax groups (see Theorems 2.1 and 2.6). These results may also be regarded as generalizations of the well-known theorem of Malcev-Baer that a radical group satisfies the maximum condition [is a polyminimax group] if all its abelian subgroups satisfy the maximum condition [are minimax groups].

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 23 , Issue 1 , February 1971 , pp. 151 - 159
Copyright
Copyright © Canadian Mathematical Society 1971

References

1. Amberg, B., Gruppen mit Minimalbedingung fur Subnormalteiler, Arch. Math. 19 (1968), 348–358.Google Scholar
2. Amberg, B., Groups with maximum conditions, Pacific J. Math. 82 (1970), 9–19.Google Scholar
3. Baer, R., Die Potenzen einer gruppentheoretischen Eigenschaft, Abh. Math. Sem. Univ. Hamburg 22 (1958), 276–294.Google Scholar
4. Baer, R., Gruppen mit Minimalbedingung, Math. Ann. 150 (1963), 1–44.Google Scholar
5. Baer, R., Noethersche Gruppen. II, Math. Ann. 165 (1966), 163–180.Google Scholar
6. Baer, R., Normalisatorreiche Gruppen, Rend. Sem. Mat. Univ. Padova 88 (1967), 358–450.Google Scholar
7. Baer, R., Polyminimaxgruppen, Math. Ann. 175 (1968), 1–43.Google Scholar
8. Plotkin, B. I., Generalized solvable and generalized nilpotent groups, Amer. Math. Soc. Transi. (2) 17 (1961), 29–115 (translation of Uspehi Mat. Nauk 18 (1958), 89-172).Google Scholar
9. Robinson, D., Infinite soluble and nilpotent groups (Queen Mary College Mathematics Notes, London, 1967).Google Scholar
10. Specht, W., Gruppentheorie (Springer-Verlag, Berlin-Göttingen-Heidelberg, 1956).Google Scholar