Download Analytic Pro-P Groups by J. D. Dixon, M. P. F. Du Sautoy, A. Mann, D. Segal PDF

By J. D. Dixon, M. P. F. Du Sautoy, A. Mann, D. Segal

ISBN-10: 0521542189

ISBN-13: 9780521542180

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Extra resources for Analytic Pro-P Groups

Example text

Clearly g contains (g); since each element of g is the z limit of a sequence of elements of (g) it follows that g p < (g). 27 Definition A group G is procyclic if G is profinite and G/N is a cyclic group for every open normal subgroup TV of G. 28 Proposition Let G be a pro-p group. Then the following are equivalent (a) G is procyclic; (b) G can be topologically generated by a one-element subset; (c) G = gZp for some g £ G; (d) G is either finite and cyclic or else is topologically isomorphic to Proof Assume that G is procyclic, and suppose that G has two distinct maximal proper open subgroups M and N.

G. G then iVM and [JV, M] are both powerfully embedded in G. (e) G has a unique maximal powerfully embedded subgroup. 2. Let M,N

Hence M = Gn+\K, and the result follows by induction. (ii) Certainly [G^, Gi] < G*+i for all i. Let n>2 and suppose inductively that [Gi,G n _i] < Gi+n-i for all i. Now we fix m > 1, and want to show that [G m ,G n ] < G m + n . Since Grn+n is closed, it will suffice to show that [Gm,Gn] < N whenever G m + n < N <\o G. Thus, in view of (i), we may replace G by the finite p-group G/N, and assume further that G m + n = 1. Then [G m ,G n _i] < G m + n _ i is central and has exponent dividing p. If g G Gm and x G G n _i we have so[G m ,G^_ 1 ] = l.

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