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The classes of finite groups with minimal sets of generators of fixed cardinalities, named B-groups, and groups with the basis property, in which every subgroup is a B-group, contain only p-groups and some {p, q}-groups. Moreover, abelian B-groups are exactly p-groups. If only generators of prime power orders are considered, then an analogue of property B is denoted by B pp and an analogue of the basis property is called the pp-basis property. These classes are larger and contain all nilpotent groups and some cyclic q-extensions of p-groups. In this paper we characterise all finite groups with the pp-basis property as products of p-groups and precisely described {p, q}-groups.2010 Mathematics subject classification: primary 20F05; secondary 20D10, 20D60.

A subset X of prime power order elements of a finite group G is called pp-independent if there is no proper subset Y of X such that Y, Φ(G) = X, Φ(G) , where Φ(G) is the Frattini subgroup of G. A group G has property B pp if all pp-independent generating sets of G have the same size. G has the pp-basis exchange property if for any pp-independent generating sets B 1 , B 2 of G and x ∈ B 1 there exists y ∈ B 2 such that (B 1 \ {x}) ∪ {y} is a ppindependent generating set of G. In this paper we describe all finite solvable groups with property B pp and all finite solvable groups with the pp-basis exchange property.

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