On Finite Groups With ‘Hidden’ Primes

Abstract

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.