Demushkin group

In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a pro-p group $G$ having a certain properties relating to duality in group cohomology. More precisely, $G$ must be such that the first cohomology group with coefficients in $\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z}$ has finite rank, the second cohomology group has rank 1, and the cup product induces a non-degenerate pairing

$$H^{1}(G,\mathbb {F} _{p})\times H^{1}(G,\mathbb {F} _{p})\rightarrow H^{2}(G,\mathbb {F} _{p}).$$

Such groups were introduced by Demuškin (1959).

Demushkin groups occur as the Galois groups of the maximal $p$-extensions of local number fields containing all $p$-th roots of unity.

References

• Demuškin, S. P. (1959), "The group of the maximum p-extension of a local field", Doklady Akademii Nauk SSSR, 128: 657–660, ISSN 0002-3264, MR 0108484
• Labute, J. (1967), "Classification of Demuskin groups", Canadian Journal of Mathematics, 19: 106–132, doi:10.4153/cjm-1967-007-8, MR 0210788
• Dummit, D.; Labute, J. (1983), "On a new characterization of Demuskin groups", Inventiones Mathematicae, 73 (3): 413–418, doi:10.1007/BF01388436, ISSN 0020-9910, MR 0718938
• Serre, Jean-Pierre (1995), "Structure de certains pro-p-groupes (d'après Demuškin)", Séminaire Bourbaki, Vol. 8, Paris: Société Mathématique de France, pp. 145–155, MR 1611538