Growth functions, p-adic analytic groups, and groups of finite coclass

Aner Shalev

doi:10.1112/jlms/s2-46.1.111

Growth functions, p-adic analytic groups, and groups of finite coclass

Aner Shalev^*

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

21 Scopus citations

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
Pages (from-to)	111-122
Number of pages	12
Journal	Journal of the London Mathematical Society
Volume	s2-46
Issue number	1
DOIs	https://doi.org/10.1112/jlms/s2-46.1.111
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

Access to Document

10.1112/jlms/s2-46.1.111

Cite this

@article{d80dec3f1dd94f2e8b43bc8a581c6c63,

title = "Growth functions, p-adic analytic groups, and groups of finite coclass",

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 pk, then G is p-adic analytic; (2) if, for some k, then G is p-acid analytic; (3) if, for some n, (G:Gn) n+[l0gP(n/2), where Gn 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.",

author = "Aner Shalev",

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",

year = "1992",

month = aug,

doi = "10.1112/jlms/s2-46.1.111",

language = "אנגלית",

volume = "s2-46",

pages = "111--122",

journal = "Journal of the London Mathematical Society",

issn = "0024-6107",

publisher = "John Wiley and Sons Ltd",

number = "1",

}

TY - JOUR

T1 - Growth functions, p-adic analytic groups, and groups of finite coclass

AU - Shalev, Aner

N1 - 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

PY - 1992/8

Y1 - 1992/8

N2 - 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 pk, then G is p-adic analytic; (2) if, for some k, then G is p-acid analytic; (3) if, for some n, (G:Gn) n+[l0gP(n/2), where Gn 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.

AB - 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 pk, then G is p-adic analytic; (2) if, for some k, then G is p-acid analytic; (3) if, for some n, (G:Gn) n+[l0gP(n/2), where Gn 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.

UR - http://www.scopus.com/inward/record.url?scp=84962991729&partnerID=8YFLogxK

U2 - 10.1112/jlms/s2-46.1.111

DO - 10.1112/jlms/s2-46.1.111

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:84962991729

SN - 0024-6107

VL - s2-46

SP - 111

EP - 122

JO - Journal of the London Mathematical Society

JF - Journal of the London Mathematical Society

IS - 1

ER -

Growth functions, p-adic analytic groups, and groups of finite coclass

Abstract

Bibliographical note

Access to Document

Other files and links

Fingerprint

Cite this