Permutation Groups, Simple Groups, and Sieve Methods

D.R. Heath-Brown, Cheryl Praeger, A. Shalev

    Research output: Contribution to journalArticle

    3 Citations (Scopus)

    Abstract

    We show that the number of integers n <= x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of A(n-1) in A(n) is similar to hx/log x, for some given constant h. This might be regarded as a noncommutative analogue of the Prime Number Theorem (which counts indices n < x of subgroups of abelian simple groups).We conclude that for most positive integers n, the only quasiprimitive permutation groups of degree n are S-n and A(n) in their natural action. This extends a similar result for primitive permutation groups obtained by Cameron, Neumann and Teague in 1982.Our proof combines group-theoretic and number-theoretic methods. In particular, we use the classification of finite simple groups, and we also apply sieve methods to estimate the size of some interesting sets of primes.
    Original languageEnglish
    Pages (from-to)347-375
    JournalIsrael Journal of Mathematics
    Volume148
    Issue number1
    DOIs
    Publication statusPublished - 2005

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