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

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**Example text**

3. The natural homomorphism F —> FΑ has kernel equal to HjveA ^ = ^-> s a v > a n < ^ ^ embeds V/K as a dense subgroup in FΑ- The group FΑ is called a profinite completion of F. ) When A consists of all the normal subgroups of finite index, FΑ is the profinite completion of F, usually denoted simply f. The kernel of F —> f is the finite residual of F. 1 Pro finite groups 19 r A = F p is the pro-p completion of F, and F embeds into Tp if and only if F is residually a finite p-group. , directed by reverse inclusion.

Ii) Deduce that if g G G and if for every N

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. ,G]] < [G,[G m ,G n _i]][G n -i,[G,G m ]] < [G,G m +n_i][G n _i,G m +i] Gm+n = 1, by the three-subgroup lemma and the inductive hypothesis. It follows that Since G is finite, this is the same as [G m , G n ] = 1, which is what we had to show. (iii) Now we assume that G is finitely generated.