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Classification of Demushkin Groups

Published online by Cambridge University Press:  20 November 2018

John P. Labute
John P. Labute*
Affiliation:
Harvard University
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A pro-p-group G is said to be a Demushkin group if

  1. (1) dimFp H1(G, Z/pZ) < ∞,

  2. (2) dimFp H2(G, Z/pZ) = 1,

  3. (3) the cup product H1(G, Z/pZ) × H1(G, Z/pZ) → H2(G, Z/pZ) is a non-degenerate bilinear form. Here FP denotes the field with p elements. If G is a Demushkin group, then G is a finitely generated topological group with n(G) = dim H1(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 ∈ F9 (F, F); cf. §1.4.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 19 , 1967 , pp. 106 - 132
Copyright
Copyright © Canadian Mathematical Society 1967

References

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