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Basic p-groups: higher commutator structure

Published online by Cambridge University Press:  09 April 2009

Paul M. Weichsel
Paul M. Weichsel
Affiliation:
University of Illinois Urbana, Illinois, U. S. A.
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The classification of groups according to the varieties they generate requires the study of a class of indecomposable elements. Such a class is the class of basic groups which have been studied in [4], [5] and [6]. A group is called basic if it is indecomposable qua group; that is, it is critical and indecomposable qua variety; that is, its variety is join-irreducible. In this note we consider the higher commutator structure of basic p-groups. Our main theme is the relation between the formal weight of the higher commutator subgroups and the class of the group. We obtain information about the power-commutator structure of a basic p-group, the kinds of laws that can hold in such a group and the varietal structure of groups of the form: Center-extended-by-X.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 16 , Issue 2 , September 1973 , pp. 200 - 206
Copyright
Copyright © Australian Mathematical Society 1973

References

[1]Hanna, Neumann, Varieties of Groups (Ergeb. Math. 37, Springer, Berlin, 1967).Google Scholar
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[3]Stewart, A. G. R., ‘On center-extended-by-metabelian groups’, Math. Ann. 185 (1970), 285–302.CrossRefGoogle Scholar
[4]Paul, M. Weichsel, ‘On critical p-groups’, Proc. London Math. Soc. (3) 14 (1964), 83–100.Google Scholar
[5]Paul, M. Weichsel, ‘Critical and basic p-groups’, Proc. Internat. Conf. Theory of Groups, Austral. Nat. Univ. Canberra, August 1965, 367–371, (Gordon and Breach, New York, 1967),Google Scholar
[6]Paul, M. Weichsel, ‘Basic p-groups’, J. Algebra 11 (1969), 331–337.Google Scholar