## Abstract

Certain growth functions associated with a pro-p group G are studied, and their impact on the structure of G is described. In particular we prove that (1) if, for some c < § and for all sufficiently large k, G has at most open subgroups of index p^{k}, then G is p-adic analytic; (2) if, for some k, then G is p-acid analytic; (3) if, for some n, (G:G_{n}) <p^{n+[l0g}_{P}(^{n/2}), where G_{n} is the lower central series of G, then G is abelian-by-finite. Results (1) to (3) sharpen previous results of Lubotzky and Mann, Lazard, Donkin and Leedham-Green respectively. Related constructions, showing that some of these results are asymptotically best possible, are analysed in some detail.

Original language | English |
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Pages (from-to) | 111-122 |

Number of pages | 12 |

Journal | Journal of the London Mathematical Society |

Volume | s2-46 |

Issue number | 1 |

DOIs | |

State | Published - Aug 1992 |

Externally published | Yes |

### Bibliographical note

Funding Information:Received 22 February 1991; revised 23 May 1991. 1991 Mathematics Subject Classification 20F14. Partially funded by an SERC fellowship, whose support is gratefully acknowledged. J. London Math. Soc. (2) 46 (1992) 111-122