Detecting pro-p-groups that are not absolute Galois groups

Galois module structure of Galois cohomology for embeddable cyclic extensions of degree p n

“…In particular, the case n = m = 1 was resolved in [13], the case m = 1 and n 1 (without the restriction ξ p ∈ E) in [11], and the case m 1 and n = 1 in [9]. As desired, these computed module structures have already led to some interesting results on the structure of absolute Galois groups: automatic realization results in [12,14], a generalization of Schreier's theorem in [6], a connection with Demuškin groups in [5], an interpretation of cohomological dimension in [8] and a characterization of certain groups that cannot appear as absolute Galois groups in [2].…”

Absolute Galois groups viewed from small quotients and the Bloch–Kato conjecture

“…These invariants are convenient for describing the F p [A/N ]-module N associated with our T -group A; see [BLMS,Section 1]. We have Proposition 4.2.…”

Groups of p-absolute Galois type that are not absolute Galois groups

“…3.3]). For this reason, the pursue of obstructions which detect effectively pro-p groups which do not occur as absolute Galois groups has great prominence in current research in Galois theory (see, e.g., [1,5,14,36]).…”

“…Let G be a pro-p group. First of all, we underline that if α ∈ H 1 (G, Z/p) is equal to 0, then the sequence (1.1) is trivially exact at both H 1 (G, Z/p) and H 2 (G, Z/p), as both cor1 N,G and res2 G,N are the identity maps, and the cup-product by α is the trivial map.The following notion was introduced in [22, Def. 6.1.1].…”

“…1 (G 1 , Z/p) → H 2 (G 1 , Z/p) denotes the cup-product by α 1 , and similarly c 2 . Since by hypothesis the sequence (1.1), with G 1 and N 1 instead of G and N , is exact at H 2 (G 1 , Z/p), and analogously with G 2 and N 2 instead of G and N , the sequence (5.3)is exact at H 2 (G 1 , Z/p) ⊕ H 2 (G 2 , Z/p) = H 2 (G, Z/p) componentwise -and thus it is exact altogether, and this concludes the proof.…”