Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-09T20:43:44.706Z Has data issue: false hasContentIssue false

Varieties of groups and isologisms

Published online by Cambridge University Press:  09 April 2009

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

Type
Research Article
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), 400407.CrossRefGoogle Scholar
[3]Bioch, J. C. and van der Waall, R. W., ‘Monomiality and isoclinism of groups’, J. Reine Angew. Math. 298 (1978), 7488.Google Scholar
[4]Gupta, C. K., ‘The free centre-by-metabelian groups’, J. Austral. Math. Soc. 16 (1973), 294299.CrossRefGoogle Scholar
[5]Gupta, C. K. and Gupta, N. D., ‘Generalized Magnus embeddings and some applications’, Math. Z. 160 (1978), 7587.CrossRefGoogle Scholar
[6]Hall, Ph., ‘The classification of prime-power groups’, J. Reine Angew. Math. 182 (1940), 130141.CrossRefGoogle Scholar
[7]Hall, Ph., ‘Verbal and marginal subgroups’, J. Reine Angew. Math. 182 (1940), 156157.CrossRefGoogle Scholar
[8]Hekster, N. S., ‘On the structure of n–isoclinism classes of groups’, J. Pure Appl. Algebra 40 (1986), 6385.CrossRefGoogle Scholar
[9]Hulse, J. A. and Lennox, J. C., ‘Marginal series in groups’, Proc. Roy. Soc. Edinburgh Sect. A 76 (1976), 139154.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), 99150.CrossRefGoogle Scholar
[12]Moghaddam, M. R. R., ‘On the Schur-Baer property’, J. Austral. Math. Soc. Ser. A 31 (1981), 343361.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), 4148.CrossRefGoogle Scholar
[19]Vermani, L. R., ‘A note on induced central extensions’, Bull. Austral. Math. Soc. 20 (1979), 411420.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), 6365.CrossRefGoogle Scholar
[22]Weichsel, P. M., ‘On critical p–groups’, Proc. London Math. Soc. (3) 14 (1964), 83100.CrossRefGoogle Scholar