Asymptotic Results for Primitive Permutation Groups

L. Pyber; A. Shalev

doi:10.1006/jabr.1996.6818

Asymptotic Results for Primitive Permutation Groups

L. Pyber^*, A. Shalev

^*Corresponding author for this work

Einstein Institute of Mathematics

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

12 Scopus citations

Abstract

We prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostn^cμ(n), where μ(n) denotes the maximal exponent occurring in the prime factorization ofn. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizerG_αof a primitive groupGof degreendoes not have the alternating group Alt(d) as a section, then the order ofGis bounded by a polynomial inn. This result extends a well-known theorem of Babai, Cameron and Pálfy. It is used to prove, for example, that ifHis a subgroup of indexnin a groupG, andHis a product ofbcyclic groups, then G:H_G≤n^cwherecdepends onb.

Original language	English
Pages (from-to)	103-124
Number of pages	22
Journal	Journal of Algebra
Volume	188
Issue number	1
DOIs	https://doi.org/10.1006/jabr.1996.6818
State	Published - 1 Feb 1997

Bibliographical note

Funding Information:
* The first author was partially supported by the Hungarian National Foundation for Scientific Research, Grant 4267.

Funding Information:
²The second author was partially supported by the Israel Science Foundation, administered by the Israel Academy of Sciences and Humanities. Author to whom correspondence should be addressed.

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10.1006/jabr.1996.6818

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abstract = "We prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostncμ(n), where μ(n) denotes the maximal exponent occurring in the prime factorization ofn. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizerGαof a primitive groupGof degreendoes not have the alternating group Alt(d) as a section, then the order ofGis bounded by a polynomial inn. This result extends a well-known theorem of Babai, Cameron and P{\'a}lfy. It is used to prove, for example, that ifHis a subgroup of indexnin a groupG, andHis a product ofbcyclic groups, then G:HG≤ncwherecdepends onb.",

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AB - We prove that the number of conjugacy classes of primitive permutation groups of degreenis at mostncμ(n), where μ(n) denotes the maximal exponent occurring in the prime factorization ofn. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizerGαof a primitive groupGof degreendoes not have the alternating group Alt(d) as a section, then the order ofGis bounded by a polynomial inn. This result extends a well-known theorem of Babai, Cameron and Pálfy. It is used to prove, for example, that ifHis a subgroup of indexnin a groupG, andHis a product ofbcyclic groups, then G:HG≤ncwherecdepends onb.

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Asymptotic Results for Primitive Permutation Groups

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