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