Demushkin group

In mathematical group theory, a Demushkin group (also written as Demuškin or Demuskin) is a [[pro-p group|pro-Template:Mvar 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 ${𝔽}_p=\mathbb{Z}/p\mathbb{Z}$ has finite rank, the second cohomology group has rank 1, and the cup product induces a non-degenerate pairing

$$H^1(G,{𝔽}_p)\times H^1(G,{𝔽}_p)\to H^2(G,{𝔽}_p).$$

Such groups were introduced by Template:Harvtxt.

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

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