Random Permutations: Some Group-Theoretic Aspects
P.J. Cameron, William M. Kantor · Cambridge University Press eBooks · 1997
The study of asymptotics of random permutations was initiated by Erdos and Tunto. in a series of papers from 1965 to 1968, and has been much studied since. Recent developments in permutation group theory make it reasonable to ask questions with a more grouptheoretic flavour. Two examples considered here are membership in a proper transitive subgroup, and the intersection of a subgroup with a random conjugate. These both arise from other topics (quasigroups, bases for permutation groups, and design constructions). 1. Permutations lying in a transitive subgroup Sn and An denote the symmetric and alternating groups on the set X = {I,.... n}. A subgroup G of S " is transitive if, for all i, j E X, there exists g E G with ig ~ j. In a preliminary version of this paper, we asked the following question: Question 1.1. Is it true that,/or almost all permutations g E Sn. the only transitive subgroups containing g are Sn and (possihly) An? Here, of course, 'almost all g E S " have property P ' means 'the proportion of elements of S " not having property P tends to 0 as n--> ex'. An affirmative answer to this question was given by Luczak and Pyber, in [15]. We will