✦ LIBER ✦

Asymptotic Results for Primitive Permutation Groups

✍ Scribed by L Pyber; A Shalev

Publisher: Elsevier Science
Year: 1997
Tongue: English
Weight: 216 KB
Volume: 188
Category: Article
ISSN: 0021-8693
DOI: 10.1006/jabr.1996.6818

No coin nor oath required. For personal study only.

✦ Synopsis

We prove that the number of conjugacy classes of primitive permutation groups cŽ n.

Ž . of degree n is at most n

, where n denotes the maximal exponent occurring in the prime factorization of n. This result is applied to investigating maximal subgroup growth of infinite groups. We then proceed by showing that if the point-stabilizer G of a primitive group G of degree n does not have the ␣ Ž . alternating group Alt d as a section, then the order of G is bounded by a polynomial in n. This result extends a well-known theorem of Babai, Cameron and Palfy. It is used to prove, for example, that if H is a subgroup of index n in a group ´< < c G , and H is a product of b cyclic groups, then G: H F n where c depends G on b.

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