Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-25T08:18:24.520Z Has data issue: false hasContentIssue false
  • English
  • Français

The subnormal coalescence of some classes of groups of finite rank

Published online by Cambridge University Press:  09 April 2009

Mark Drukker ,
Derek J. S. Robinson  and
Ian Stewart
Mark Drukker
Affiliation:
University of WarwickCoventry, Warwickshire Cv4, 7AL England
Derek J. S. Robinson
Affiliation:
University of WarwickCoventry, Warwickshire Cv4, 7AL England
Ian Stewart
Affiliation:
University of IllinoisUrbana, Illinois 61801, U.S.A.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A class of groups forms a (subnormal) coalition class, or is (subnormally) coalescent, if whenever H and K are subnormal -subgroups of a group G then their join <H, K> is also a subnormal -subgroup of G. Among the known coalition classes are those of finite groups and polycylic groups (Wielandt [15]); groups with maximal condition for subgroups (Baer [1]); finitely generated nilpotent groups (Baer [2]); groups with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 3 , November 1973 , pp. 324 - 327
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Baer, R., ‘Lokal Noethersche Gruppen’, Math. Z. 66 (1957), 341–363.CrossRefGoogle Scholar
[2]Baer, R., ‘Nil-Gruppen’, Math. Z. 62 (1955), 402–437.CrossRefGoogle Scholar
[3]Baer, R., ‘Polyminimaxgruppen’, Math. Ann. 175 (1968), 1–43.CrossRefGoogle Scholar
[4]Hirsch, K. A., ‘Über lokal-nilpotente Gruppen’, Math. Z. 63 (1955), 290–294.CrossRefGoogle Scholar
[5]Kuro`, A. G., Theory of groups vol. 2, (Chelsea, New York, 1956). Translated by Hirsch, K. A..Google Scholar
[6]Mal'cev, A.I., ‘On certain classes of infinite soluble groups’, Math. Sb. (N. S.) 28 (70) (1951), 567–588.Google Scholar
Amer. Math. Soc. Translations (2) 2 (1956), 1–21.CrossRefGoogle Scholar
[7]Plotkin, B. I., ‘On some criteria of locally nilponent groups’, Uspehi Mat. Nauk (N.S.) 9 (1954), 181–186.Google Scholar
Amer. Math. Soc. Translations (2) (17 1961), 1–8.CrossRefGoogle Scholar
[8]Robinson, D. J. S., ‘On the theory of subnormal subgroups’, Math, Z. 89 (1965), 30–51.CrossRefGoogle Scholar
[9]Robinson, D. J. S., Infinite soluble and nilpotent groups, (QMC Mathematics Notes, London 1968).Google Scholar
[10]Robinson, D. J. S., ‘A note on groups of finite rank’, Compositio Math. 21 (1969), 240–246.Google Scholar
[11]Roseblade, J. E., ‘On certain subnormal coalition classes’, J. Algebra I (1964), 132–138.CrossRefGoogle Scholar
[12]Roseblade, J. E., ‘A note on subnormal coalition classes’, Math. Z. 90 (1965), 373–375.CrossRefGoogle Scholar
[13]Roseblade, J. E. and Stonehewer, S. E., ‘Subjunctive and locally coalescent classes’, J. Algebra 8 (1968), 423–435.CrossRefGoogle Scholar
[14]Stonehewer, S. E., ‘The join of finitely many subnormal subgroups’, Bull. London. Math. Soc. 2 (1970), 77–82.CrossRefGoogle Scholar
[15]Wielandt, H., ‘Eine Verallgemeinerung der invarianten Untergruppen’, Math. Z. 45 (1939), 209–244.CrossRefGoogle Scholar