Let k be an algebraic number field and A a finite, irreducible Gk-module. We show that the localization mapping H1(k,A)→∏pH1(kp,A) is injective (or that for imbedding problems with such a kernel A the full local-global principle holds), if the trivializing extension k(A)/k is p-solvable (p is the prime number with pA=0), and that for nonsolvable k(A)/k this is in general false (counterexample already for Gal(k(A)/k)=A5). This answers a question of O. Neumann [Math. Nachr. 72 (1976), 305–320; MR0417124 (54 #5184)].