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 , generated by *n*th 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 , then there is a function such that if every word in of length 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