Groups whose subgroup growth is less than linear

Aner Shalev

doi:10.1142/S0218196797000071

Groups whose subgroup growth is less than linear

Aner Shalev^*

^*Corresponding author for this work

Einstein Institute of Mathematics

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

7 Scopus citations

Abstract

Let G be a residually finite group and let a_n(G) denote the number of index n subgroups of G. It is shown that a_n(G)/n → 0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.

Original language	English
Pages (from-to)	77-91
Number of pages	15
Journal	International Journal of Algebra and Computation
Volume	7
Issue number	1
DOIs	https://doi.org/10.1142/S0218196797000071
State	Published - Feb 1997

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abstract = "Let G be a residually finite group and let an(G) denote the number of index n subgroups of G. It is shown that an(G)/n → 0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.",

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AB - Let G be a residually finite group and let an(G) denote the number of index n subgroups of G. It is shown that an(G)/n → 0 if and only if G has a finite index central subgroup whose finite quotients are all cyclic. As an application we show that the degree of a group of polynomial subgroup growth cannot lie strictly between 0 and 1.

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JO - International Journal of Algebra and Computation

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Groups whose subgroup growth is less than linear

Abstract

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