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Closed Subgroups of Profinite Groups
Author(s) -
Segal Dan
Publication year - 2000
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s002461150001234x
Subject(s) - mathematics , profinite group , pure mathematics , mathematics education , group (periodic table) , organic chemistry , chemistry
Theorem 1 asserts that in a finitely generated prosoluble group, every subgroup of finite index is open . This generalises an old result of Serre about pro‐ p groups. It follows by a standard argument from Theorem 2: in a d‐generator finite soluble group, every element of the derived group is equal to a product of 72d 2 + 46d commutators . This result about finite soluble groups is proved by induction on the order of the group, and is elementary though rather intricate. The essence of the proof lies in reducing the problem to one about the number of solutions of quadratic equations over a finite field. Corollaries include the following. Let Γ be a finitely generated prosoluble group. Then each term of the lower central series of Γ and each power subgroup Γ n is closed . 1991 Mathematics Subject Classification : 20E18, 20D10.