Let K/k be a finite Galois CM-extension of number fields whose Galois group G is monomial and S a finite set of places of k. Then the "Stickelberger element" θ K/k,S is defined. Concerning this element, Andreas Nickel formulated the non-abelian Brumer and Brumer-Stark conjectures and their "weak" versions. In this paper, when G is a monomial group, we prove that the weak non-abelian conjectures are reduced to the weak conjectures for abelian subextensions. We write D4p, Q 2 n+2 and A4 for the dihedral group of order 4p for any odd prime p, the generalized quaternion group of order 2 n+2 for any natural number n and the alternating group on 4 letters respectively. Suppose that G is isomorphic to D4p, Q 2 n+2 or A4 ×Z/2Z. Then we prove the l-parts of the weak non-abelian conjectures, where l = 2 in the quaternion case, and l is an arbitrary prime which does not split in Q(ζp) in the dihedral case and in Q(ζ3) in the alternating case. In particular, we do not exclude the 2-part of the conjectures and do not assume that S contains all finite places which ramify in K/k in contrast with Nickel's formulation.

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In this paper, we prove, for primes l satisfying some conditions, the l-parts of the Galois Brumer-Stark conjecture, which is formulated by Dejou and Roblot for Galois CM-extensions with dihedral or generalized quaternion Galois group of specified degrees.

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