Hostname: page-component-7479d7b7d-qlrfm Total loading time: 0 Render date: 2024-07-11T03:32:03.206Z Has data issue: false hasContentIssue false
  • English
  • Français

On the Schur-Baer property

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups

Published online by Cambridge University Press:  09 April 2009

Mohammad Reza R. Moghaddam
Mohammad Reza R. Moghaddam
Affiliation:
Department of Mathematics, Faculty of Science, University of Mashhad, Mashhad, Iran
Rights & Permissions [Opens in a new window]

Abstract

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.

In 1957 P. Hall conjectured that every (finitely based) variety has the property that, for every group G, if the marginal factor-group is finite, then the verbal subgroup is also finite. The content of this paper is to present a precise bound for the order of the verbal subgroup of a G when the marginal factor-group is of order Pn (p a prime and n > 1) with respect to the variety of polynilpotent groups of a given class row. We also construct an example to show that the bound is attained and furthermore, we obtain a bound for the order of the Baer-invariant of a finite p-group with respect to the variety of polynilpotent groups.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups 20F12: Commutator calculus
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 31 , Issue 3 , October 1981 , pp. 343 - 361
Copyright
Copyright © Australian Mathematical Society 1981

References

Green, J. A. (1956), ‘On the number of automorphisms of a finite group’, Proc. Roy. Soc. London Ser. A 237, 574–581.Google Scholar
Hulse, J. A. and Lennox, J. C. (1976), ‘Marginal series in groups’, Proc. Roy. Soc. Edinburgh, 67 Sect. A, 139–154.Google Scholar
Hall, M. (1959), The theory of groups (Macmillan, New York).Google Scholar
Hall, P. (1957), Nilpotent groups (Canad. Math. Cong. Univ. Alberta.) (Queen Mary College Math. Notes, (1970)).Google Scholar
Jones, M. R. (1972), ‘Multiplicators of p-groups’, Math. Z. 127, 165–166.Google Scholar
Leedham-Green, C. R. and McKay, S. (1976), ‘Baer-invariants, isologism, varietal laws and homology, Acta Math. 137, 99–150.CrossRefGoogle Scholar
Moghaddam, M. R. R. (1975), A varietal generalisation of the Schur multiplicator (Ph.D. Thesis, University of London).Google Scholar
Moghaddam, M. R. R. (1979), ‘The Baer-invariant of a direct product’, Arch. Math. (Basel) 33, 504–511.Google Scholar
Neumann, H. (1967), Varieties of groups (Springer, Berlin).Google Scholar
Robinson, D. J. S. (1972), Finiteness conditions and generalized soluble groups, Part 1 (Springer, Berlin).Google Scholar
Schur, I. (1907), ‘Untersuchungen über die Darstellung der endlichen Gruppen durch gebrochene lineare Subtitutionen’, J. Reine Angew. Math. 132, 85–137.Google Scholar
Turner-Smith, R. F. (1964), ‘Marginal subgroup properties for outer commutator words’, Proc. London Math. Soc. 14, 321–341.Google Scholar
Wiegold, J. (1965), ‘Multiplicators and groups with finite central factor-group’, Math. Z. 89, 345–347.Google Scholar