Permutations contained in transitive subgroups

Sean Eberhard; Kevin Ford; Dimitris Koukoulopoulos

doi:10.19086/da.849

Permutations contained in transitive subgroups

Sean Eberhard, Kevin Ford, Dimitris Koukoulopoulos

Research output: Contribution to journal › Article › peer-review

7 Citations (Scopus)

9 Downloads (Pure)

Abstract

In the first paper in this series we estimated the probability that a random permutation π ∈ S_n has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that π has m disjoint fixed sets of prescribed sizes k₁,..,k_m, where k₁+···+k_m = n. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than S_n or A_n. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estimate for the proportion of permutations contained in a primitive subgroup other than S_n or A_n.

Original language	English
Number of pages	36
Journal	Discrete Analysis
Volume	2016
Issue number	12
DOIs	https://doi.org/10.19086/da.849
Publication status	Published - 29 Jul 2016
Externally published	Yes

Keywords

Primitive groups
Transitive groups
Łuczak-Pyber theorem

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.19086/da.849Licence: CC BY

Permutations contained in transitive subgroups
Copyright 2016 The Authors. This is an open access article published under a Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium, provided the author and source are cited.
Final published version, 493 KBLicence: CC BY

Cite this

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AU - Ford, Kevin

AU - Koukoulopoulos, Dimitris

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N2 - In the first paper in this series we estimated the probability that a random permutation π ∈ Sn has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that π has m disjoint fixed sets of prescribed sizes k1,..,km, where k1+···+km = n. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than Sn or An. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estimate for the proportion of permutations contained in a primitive subgroup other than Sn or An.

AB - In the first paper in this series we estimated the probability that a random permutation π ∈ Sn has a fixed set of a given size. In this paper, we elaborate on the same method to estimate the probability that π has m disjoint fixed sets of prescribed sizes k1,..,km, where k1+···+km = n. We deduce an estimate for the proportion of permutations contained in a transitive subgroup other than Sn or An. This theorem consists of two parts: an estimate for the proportion of permutations contained in an imprimitive transitive subgroup, and an estimate for the proportion of permutations contained in a primitive subgroup other than Sn or An.

KW - Primitive groups

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KW - Łuczak-Pyber theorem

U2 - 10.19086/da.849

DO - 10.19086/da.849

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SN - 2397-3129

VL - 2016

JO - Discrete Analysis

JF - Discrete Analysis

IS - 12

ER -

Permutations contained in transitive subgroups

Abstract

Keywords

ASJC Scopus subject areas

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