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

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X s ) the element x'k = wxk has order p^. Show that then (x±,... , x s , x^ + 1 ,... ] 10. Let M = Gpn if pis odd, M = (G 2")2 if p = 2. Suppose that M is not powerful. Show that d(M) > pn. B is elementary abelian of rank > p n , and (H) [A, xpU] jt 5 . 12 and consider the exponent of Ur(Fp). ] 11. The wreath product Cp I X, for a finite group X, is the semidirect product of the group algebra FP[X] by X, with X acting by right multiplication. Now let H be a finite p-group containing an elementary abelian normal subgroup A and an element x such that [>l,xp ] ^ 1.

X^ G G such that for each N <\o G, (x±N,... 3), and part (i) shows that { # 1 , . . ,Xd} generates H topologically. 6 Proposition If G is a finitely generated profinite group and m is a positive integer then G has only finitely many open subgroups of index m, and every open subgroup contains an open topologically characteristic subgroup. ) Proof Suppose G is topologically generated by a d-element subset. Then there are at most (m\)d continuous homomorphisms of G into the symmetric group 5 m (where Sm has the discrete topology).

5. Suppose G is powerful and has exponent pe. For each k, let E^ = i> x p is an {x £ G \ xp = 1 } . (a) Show that the mapping x — endomorphism of G. (6) Show that JKe-i is a subgroup of G, of order equal to \G : P e (G)|. (c) Show by an example that Ee-\ need not be powerful, (d) Show that if all characteristic subgroups of G are powerful then Ek is a subgroup of G for each k. 6. ) = $($*(£)). Let 5 = su Pi > o d($*(G)) and put W = $ A ( s )+ e (G), where e = 0 if p is odd, e = 1 if p = 2. Show that W is powerful and that \G:W\< p s ( A ( 5 )+ e ).