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The subnormal coalescence of some classes of groups of finite rank

  • Mark Drukker (a1), Derek J. S. Robinson (a1) and Ian Stewart (a2)
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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with maximal or minimal condition on subnormal subgroups (Robinson [8], Roseblade [11, 12]); minimax groups (Roseblade, unpublished); and any subjunctive class of finitely generated groups (Roseblade and Stonehewer [13]).

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References
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[1]Baer, R., ‘Lokal Noethersche Gruppen’, Math. Z. 66 (1957), 341363.
[2]Baer, R., ‘Nil-Gruppen’, Math. Z. 62 (1955), 402437.
[3]Baer, R., ‘Polyminimaxgruppen’, Math. Ann. 175 (1968), 143.
[4]Hirsch, K. A., ‘Über lokal-nilpotente Gruppen’, Math. Z. 63 (1955), 290294.
[5]Kuro`, A. G., Theory of groups vol. 2, (Chelsea, New York, 1956). Translated by Hirsch, K. A..
[6]Mal'cev, A.I., ‘On certain classes of infinite soluble groups’, Math. Sb. (N. S.) 28 (70) (1951), 567588.
Amer. Math. Soc. Translations (2) 2 (1956), 121.
[7]Plotkin, B. I., ‘On some criteria of locally nilponent groups’, Uspehi Mat. Nauk (N.S.) 9 (1954), 181186.
Amer. Math. Soc. Translations (2) (17 1961), 18.
[8]Robinson, D. J. S., ‘On the theory of subnormal subgroups’, Math, Z. 89 (1965), 3051.
[9]Robinson, D. J. S., Infinite soluble and nilpotent groups, (QMC Mathematics Notes, London 1968).
[10]Robinson, D. J. S., ‘A note on groups of finite rank’, Compositio Math. 21 (1969), 240246.
[11]Roseblade, J. E., ‘On certain subnormal coalition classes’, J. Algebra I (1964), 132138.
[12]Roseblade, J. E., ‘A note on subnormal coalition classes’, Math. Z. 90 (1965), 373375.
[13]Roseblade, J. E. and Stonehewer, S. E., ‘Subjunctive and locally coalescent classes’, J. Algebra 8 (1968), 423435.
[14]Stonehewer, S. E., ‘The join of finitely many subnormal subgroups’, Bull. London. Math. Soc. 2 (1970), 7782.
[15]Wielandt, H., ‘Eine Verallgemeinerung der invarianten Untergruppen’, Math. Z. 45 (1939), 209244.
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Journal of the Australian Mathematical Society
  • ISSN: 1446-7887
  • EISSN: 1446-8107
  • URL: /core/journals/journal-of-the-australian-mathematical-society
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