Classification of Demushkin Groups
John P. Labute
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Source: Crossref
Published: Jan 1, 1967
DOI: 10.4153/cjm-1967-007-8
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A pro- p -group G is said to be a Demushkin group if (1) dim F p H 1 ( G , Z/ p Z) < ∞, (2) dim F p H 2 ( G , Z/ p Z) = 1, (3) the cup product H 1 (G, Z/pZ) × H 1 (G, Z/pZ) → H 2 (G, Z/pZ) is a non-degenerate bilinear form. Here F P denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H 1 (G, Z/pZ) as the minimal number of topological generators; cf. §1.3. Condition (2) means that there is only one relation among a minimal system of generators for G; that is, G is isomorphic to a quotient F/(r), where F is a free pro- p -group of rank n = n(G) and (r) is the closed normal subgroup of F generated by an element r ∈ F 9 (F, F); cf. §1.4.
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