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On Finite Groups With ‘Hidden’ Primes

Published online by Cambridge University Press:  09 April 2009

L. G. Kovács ,
Joachim Neubüser  and
B. H. Neumann
L. G. Kovács
Affiliation:
Australian National University, Canberra
Joachim Neubüser
Affiliation:
Rheinisch-Westfälische Technische Hochschule, Aachen
B. H. Neumann
Affiliation:
Australian National University, Canberra
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The starting point of this investigation was a question put to us by Martin B. Powell: If the prime number p divides the order of the finite group G, must there be a minimal set of generators of G that contains an element whose order is divisible by p? A set of generators of G is minimal if no set with fewer elements generates G. A minimal set of generators is clearly irredundant, in the sense that no proper subset of it generates G; an irredundant set of generators, however, need not be minimal, as is easily seen from the example of a cyclic group of composite (or infinite) order. Powell's question can be asked for irredundant instead of minimal sets of generators; it turns out that the answer is not the same in these two cases. A different formulation, together with some notation, may make the situation clearer.

Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 12 , Issue 3 , August 1971 , pp. 287 - 300
Copyright
Copyright © Australian Mathematical Society 1971

References

[1]Curtis, Charles W., Reiner, Irving, Representation theory of finite groups and associative algebras (Interscience, New York, London, Sydney, 1962).Google Scholar
[2]Frattini, G., ‘Intorno alla generazione dei gruppi di operazioni’, Atti R. Accad. dei Lincei, Rendiconti (IV) 1 (1885), 281–285.Google Scholar
[3]Gaschütz, Wolfgang, ‘Über die Φ-Untergruppe endlicher Gruppen’, Math. Zeitschr. 58 (1953), 160–170.CrossRefGoogle Scholar
[4]Gaschütz, Wolfgang, ‘Zu einem von B. H. und H. Neumann gesteilten Problem’, Math. Nachr. 14 (1956), 249–252.CrossRefGoogle Scholar
[5]Hall, P., ‘The Eulerian functions of a group’, Quart. J. Math. (Oxford) 7 (1936), 134–151.CrossRefGoogle Scholar
[6]Huppert, B., Endliche Gruppen I (Springer-Verlag, Berlin, Heidelberg, New York, 1967).CrossRefGoogle Scholar
[7]Neumann, B. H., ‘Twisted wreath products of groups’, Arch. Math. 14 (1963), 1–6.CrossRefGoogle Scholar
[8]Neumann, Bernhard H. und Neumann, Hanna, ‘Zwei Klassen charakteristischer Untergruppen und ihre Faktorgruppen’, Math. Nachr. 4 (1951), 106–125.Google Scholar
[9]Scott, W. R., Group theory (Prentice-Hall, Englewood Cliffs, 1964).Google Scholar
[10]Zassenhaus, Hans, ‘Über endliche Fastkörper’, Abh. math. Sem. Hansisch. Univ. 11 (1936), 187–220.CrossRefGoogle Scholar