Closed Subgroups of Profinite Groups |
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Authors: | Segal Dan |
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Institution: | All Souls College Oxford, OX1 4AL, UK; dan.segal{at}all-souls.ox.ac.uk |
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Abstract: | Theorem 1 asserts that in a finitely generated prosoluble group,every subgroup of finite index is open. This generalises anold result of Serre about pro-p groups. It follows by a standardargument from Theorem 2: in a d-generator finite soluble group,every element of the derived group is equal to a product of72d2 + 46d commutators. This result about finite soluble groupsis proved by induction on the order of the group, and is elementarythough rather intricate. The essence of the proof lies in reducingthe problem to one about the number of solutions of quadraticequations over a finite field. Corollaries include the following.Let be a finitely generated prosoluble group. Then each termof the lower central series of and each power subgroup n isclosed. 1991 Mathematics Subject Classification: 20E18, 20D10. |
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Keywords: | finite index subgroups derived group products of commutators |
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