A generalization of the Burnside basis theorem

A BB-group is a group such that all its minimal generating sets (with respect to inclusion) have the same size. We prove that the class of finite BB-groups is closed under taking quotients and that every finite BB-group is solvable. Via a complete classification of Frattini-free finite BB-groups we obtain a general structure theorem for finite BB-groups. Applications include new proofs for the characterization of finite matroid groups.