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Transactions of the American Mathematical Society

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On power subgroups of profinite groups


Author: Consuelo Martínez
Journal: Trans. Amer. Math. Soc. 345 (1994), 865-869
MSC: Primary 20E18
DOI: https://doi.org/10.1090/S0002-9947-1994-1264149-8
MathSciNet review: 1264149
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Abstract: In this paper we prove that if G is a finitely generated pro-(finite nilpotent) group, then every subgroup $ {G^n}$, generated by nth powers of elements of G, is closed in G. It is also obtained, as a consequence of the above proof, that if G is a nilpotent group generated by m elements $ {x_1}, \ldots ,{x_m}$, then there is a function $ f(m,n)$ such that if every word in $ x_i^{ \pm 1}$ of length $ \leq f(m,n)$ has order n, then G is a group of exponent n. This question had been formulated by Ol'shansky in the general case and, in this paper, is proved in the solvable case and the problem is reduced to the existence of such function for finite simple groups.


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DOI: https://doi.org/10.1090/S0002-9947-1994-1264149-8
Article copyright: © Copyright 1994 American Mathematical Society