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Commutator subgroups of finite p-groups

To Bernhard Hermann Neumann on his 60th birthday

Published online by Cambridge University Press:  09 April 2009

James Wiegold
James Wiegold
Affiliation:
Australian National University, University College, Cardiff
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Within five minutes of the start of my first meeting with Bernhard Neumann as his research student, late in 1954, he suggested the following problem to me. Let G be a group in which the cardinals of the classes of conjugate elements are boundedly finite with maximum η, say. Then the commutator subgroup G′ is finite [6], Is the order |G′| of G′ bounded in terms of n? I distinctly recall these words of Neumann: “That should provide us with a start, I think”. He was right: more than just a start, the problem has been a continuing stimulus to a study of questions in fields as far apart as permutation group theory ([7], [11], [12], [14], and some unpublished work of Peter M. Neumann) and multiplicator theory ([13], [2]), as well as attracting interest in its own right.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 10 , Issue 3-4 , November 1969 , pp. 480 - 484
Copyright
Copyright © Australian Mathematical Society 1969

References

[1]Bride, I. M., Ph. D. Thesis, University of Manchester, 1968.Google Scholar
[2]Gaschütz, W., Neubüser, J. and Ti, Yen, ‘Über den Multiplikator von þ-Gruppen’, Math. Zeitschr. 100, (1967) 93–96.CrossRefGoogle Scholar
[3]Knoche, H.-G., ‘Über den Frobenius'schen Klassenbegriff in nilpotenten Gruppen’, Math. Zeitschr. 55, (1951) 71–83.CrossRefGoogle Scholar
[4]Leedham-Green, C. R., Peter, M.Neumann, and James, Wiegold, ‘The breadth and the class of a finite þ-group’, to be published in J. London Math. Soc.Google Scholar
[5]Macdonald, I. D., ‘Some explicit bounds in groups with finite derived groups’, Proc. London Math. Soc. (3) 11 (1961), 23–56.CrossRefGoogle Scholar
[6]Neumann, B. H., ‘Groups covered by permutable subsets’, J. London Math. Soc. 29 (1954), 236–248.Google Scholar
[7]Peter, M. Neumann, D. Phil. Thesis, University of Oxford, 1966.Google Scholar
[8]Peter, M. Neumann, ‘An improved bound for BFC þ-groups’, to be published in J. Aust. Math. Soc.Google Scholar
[9]Schur, I., ‘Über die Darstellung der endlichen Gruppen durch gebrochene lineare Substitutionen’, J. reine angew. Math. 127 (1904) 20–50.Google Scholar
[10]James, Wiegold, M. Sc. Thesis, University of Manchester, 1955.Google Scholar
[11]James, Wiegold, ‘Groups with boundedly finite classes of conjugate elements’, Proc. Roy. Soc. (A) 238 (1956), 389–401.Google Scholar
[12]James, Wiegold, ‘Transitive subgroups of transitive þ-groups’, Math. Zeitschr. 96 (1967) 294–295.Google Scholar
[13]James, Wiegold, ‘Multiplicators and groups with finite central factor-groups’, Math. Zeitschr. 89 (1965) 345–347.Google Scholar
[14]Shepperd, J. A. H. and James, Wiegold, ‘Transitive permutation groups and groups with finite derived groups’, Math. Zeitschr. 81 (1963), 279–285.CrossRefGoogle Scholar