On finite groups whose derived subgroup has bounded rank

الأصل · EN

Let G be a finite group with derived subgroup of rank r. We prove that ≤ |G'|²ʳ. Motivated by the results of I. M. Isaacs in isa we show that if G is capable then ≤ |G'|⁴ʳ. This answers a question of L. Pyber. We prove that if G is a capable p-group then the rank of G/Z(G) is bounded above in terms of the rank of G'.