Independent sets of generators of prime power order

Abstract A subset X of a finite group G is said to be prime-power-independent if each element in X has prime power order and there is no proper subset Y of X with 〈 Y , Φ ( G ) 〉 = 〈 X , Φ ( G ) 〉 , where Φ ( G ) is the Frattini subgroup of G . A group G is B p p if all prime-power-independent generating sets for G have the same cardinality. We prove that, if G is B p p , then G is solvable. Pivoting on some recent results of Krempa and Stocka (2014); Stocka (2020), this yields a complete classification of B p p -groups.