Under r-generation conditions on commutator subsets in finite groups, [G,P] or G' has r-bounded rank, with p-solubility or extra conditions required.
Lucchini, A bound on the number of generators of a finite gro up, Arch.Math
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Rank type conditions on commutators in finite groups
Under r-generation conditions on commutator subsets in finite groups, [G,P] or G' has r-bounded rank, with p-solubility or extra conditions required.