In this article we introduce some new invariants for blocks of groups algebras of finite groups which are based on the notion of complexity as defined by Alperin. These invariants are used to obtain some contributions to a problem posed by Scott and Alperin in 1986 : the isomorphism type of defect groups of a block is determined by the block algebra (i.e. independently of the chosen basis) for certain types of defect groups. For example, thsi applies to the situation where the defect groups are known to be abelian or, more generally, hamiltonian. Our result hold for blocks of integral as well as modular group algebras.