Permutation groups, simple groups, and Sieve methods

Journal article

We show that the number of integers n ≤ x which occur as indices of subgroups of nonabelian finite simple groups, excluding that of An-1 in An, is ∼ 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 Sn and An 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.

Authors: Heath-Brown, D; Praeger, C; Shalev, A

• Publication status: Published
• Journal: ISRAEL JOURNAL OF MATHEMATICS
• Volume: 148
• Issue: 1
• Pages: 347-375
• Publication date: 2005-01-01
• DOI: 10.1007/BF02775443
• EISSN: 1565-8511
• ISSN: 0021-2172
• Language: English