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

ISBN-10: 0521542189

ISBN-13: 9780521542180

ISBN-10: 0521650119

ISBN-13: 9780521650113

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G/K. e. e. G/K. e. G and x € G then (N, x) is powerful. (iv) // N is not powerfully embedded in W, then there exists a normal subgroup J of G such that • if p is odd, NP[N, W, W] < J < Np[N, W) 37 and \NP[N, W] : J\ = p; 38 Powerful p-groups • if p = 2, NA[N, W]2[N, W, W] < J < N4[N, W] and \NA[N, W] : J\ = 2. Proof Parts (i) and (ii) are obvious from the definition. To prove (iii), put H = (N,x). e. G then [H,H] < Np < Hp (respectively, [H,H] < H4 if p = 2). For part (iv), suppose that p is odd and that [AT, W] ^ Np.

2 Proposition Let G be a profinite group. (i) Every open subgroup of G is closed, has finite index in G, and contains an open normal subgroup of G. A closed subgroup of G is 15 16 Pro finite groups and pro-p groups open if and only if it has finite index. The family of all open subgroups of G intersects in {1}. (ii) A subset of G is open if and only if it is a union of cosets of open normal subgroups. (iii) For any subset X of G, x= N

That there exists a unique 6 making the diagram H 3 \ • > G H or Hp commute, where j is the natural map H —• H (or H —» Hp). Deduce that if if is a dense subgroup of G then G is a (continuous) epimorphic image of H (respectively Hp). (iii) Give an example of a finitely generated prop group G and a dense subgroup if of G, with H finitely generated as an abstract group, such that Hp ¥ G. 4. Let r be a group, A a family of normal subgroups of finite index in r , directed with respect to reverse inclusion, and put G = t \ .

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

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