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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

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