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Varieties of groups and isologisms

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

Published online by Cambridge University Press:  09 April 2009

N. S. Hekster
N. S. Hekster
Affiliation:
Mathematisch Instituut, University van AmsterdamRoetersstraat 15, 1018 WB Amsterdam, The Netherlands
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Abstract

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In order to classify solvable groups Philip Hall introduced in 1939 the concept of isoclinism. Subsequently he defined a more general notion called isologism. This is so to speak isoclinism with respect to a certain variety of groups. The equivalence relation isologism partitions the class of all groups into families. The present paper is concerned with the internal structure of these families.

Keywords

variety of groupsisologism.

MSC classification

Secondary: 20E10: Quasivarieties and varieties of groups 20E99: None of the above, but in this section 20F14: Derived series, central series, and generalizations
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 46 , Issue 1 , February 1989 , pp. 22 - 60
Copyright
Copyright © Australian Mathematical Society 1989

References

[1]Beyl, F. R. and Tappe, J., Group extensions, representations and the Schur multiplicator, (Lecture Notes in Math., vol. 958, Springer, Berlin-Heidelberg-New York, 1982).CrossRefGoogle Scholar
[2]Bioch, J. C., ‘On n-isoclinic groups’, Indag. Math. 38 (1976), 400–407.CrossRefGoogle Scholar
[3]Bioch, J. C. and van der Waall, R. W., ‘Monomiality and isoclinism of groups’, J. Reine Angew. Math. 298 (1978), 74–88.Google Scholar
[4]Gupta, C. K., ‘The free centre-by-metabelian groups’, J. Austral. Math. Soc. 16 (1973), 294–299.CrossRefGoogle Scholar
[5]Gupta, C. K. and Gupta, N. D., ‘Generalized Magnus embeddings and some applications’, Math. Z. 160 (1978), 75–87.CrossRefGoogle Scholar
[6]Hall, Ph., ‘The classification of prime-power groups’, J. Reine Angew. Math. 182 (1940), 130–141.CrossRefGoogle Scholar
[7]Hall, Ph., ‘Verbal and marginal subgroups’, J. Reine Angew. Math. 182 (1940), 156–157.CrossRefGoogle Scholar
[8]Hekster, N. S., ‘On the structure of n–isoclinism classes of groups’, J. Pure Appl. Algebra 40 (1986), 63–85.CrossRefGoogle Scholar
[9]Hulse, J. A. and Lennox, J. C., ‘Marginal series in groups’, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 139–154.CrossRefGoogle Scholar
[10]Huppert, B., Endliche Gruppen I (Springer, Berlin-Heidelberg-New York, 1979).Google Scholar
[11]Leedham-Green, C. R. and McKay, S., ‘Baer-invariants, isologism, varietal laws and homology’, Acta Math. 137 (1976), 99–150.CrossRefGoogle Scholar
[12]Moghaddam, M. R. R., ‘On the Schur-Baer property’, J. Austral. Math. Soc. Ser. A 31 (1981), 343–361.CrossRefGoogle Scholar
[13]Neumann, H., Varieties of groups, (Ergebnisse der Math., Neue Folge 37, Springer, Berlin, 1967).CrossRefGoogle Scholar
[14]Robinson, D. J. S., Finiteness conditions and generalized soluble groups, (Ergebnisse der Math., Neue Folge 62, 63, Springer, Berlin, 1972).CrossRefGoogle Scholar
[15]Robinson, D. J. S., A course in the theory of groups, (Graduate Texts in Math. 80, Springer, Berlin-Heidelberg-New York, 1982).CrossRefGoogle Scholar
[16]Segal, D., Polycyclic groups (Cambridge Tracts in Math. 82, Cambridge Univ. Press, 1983).CrossRefGoogle Scholar
[17]Stammbach, U., Homology in group theory, (Lecture Notes in Math. 159, Springer, Berlin-Heidelberg-New York, 1973).CrossRefGoogle Scholar
[18]Stroud, P. W., ‘On a property of verbal and marginal subgroups’, Proc. Cambridge Philos. Soc. 61 (1965), 41–48.CrossRefGoogle Scholar
[19]Vermani, L. R., ‘A note on induced central extensions’, Bull. Austral. Math. Soc. 20 (1979), 411–420.CrossRefGoogle Scholar
[20]van der Waall, R. W., ‘On n–isoclinic embedding of groups’, to appear in J. Pure Appl. Algebra.Google Scholar
[21]Weichsel, P. M., ‘On isoclinism’, J. London Math. Soc. 38 (1963), 63–65.CrossRefGoogle Scholar
[22]Weichsel, P. M., ‘On critical p–groups’, Proc. London Math. Soc. (3) 14 (1964), 83–100.CrossRefGoogle Scholar