Identifying long cycles in finite alternating and symmetric groups acting on subsets

Stephen Alexander Linton; Alice C. Niemeyer; Cheryl E. Praeger

doi:10.13069/jacodesmath.28239

Identifying long cycles in finite alternating and symmetric groups acting on subsets

Stephen Alexander Linton, Alice C. Niemeyer, Cheryl E. Praeger

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

Abstract

Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group A_n or S_n acting on the k-element subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n − 1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.

Original language	English
Pages (from-to)	117-149
Journal	Journal of Algebra Combinatorics Discrete Structures and Applications
Volume	2
Issue number	2
DOIs	https://doi.org/10.13069/jacodesmath.28239
Publication status	Published - May 2015

Keywords

Symmetric and alternating groups in subset actions
Large base permutation groups
Finding long cycles

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10.13069/jacodesmath.28239

JACODESMATH-2015-2-2.744
Copyright Journal of Algebra Combinatorics Discrete Structures and Applications 2015. This is the version of record of the work posted here by permission of Journal of Algebra Combinatorics Discrete Structures and Applications.
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EP/C523229/1: Multidisciplinary Critical Mass in Computational Algebra and Applications
Linton, S. A. (PI), Gent, I. P. (CoI), Leonhardt, U. (CoI), Mackenzie, A. (CoI), Miguel, I. J. (CoI), Quick, M. (CoI) & Ruskuc, N. (CoI)
EPSRC
1/09/05 → 31/08/10
Project: Standard

Cite this

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title = "Identifying long cycles in finite alternating and symmetric groups acting on subsets",

abstract = "Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the k-element subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n − 1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.",

keywords = "Symmetric and alternating groups in subset actions, Large base permutation groups, Finding long cycles",

author = "Linton, {Stephen Alexander} and Niemeyer, {Alice C.} and Praeger, {Cheryl E.}",

year = "2015",

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AU - Linton, Stephen Alexander

AU - Niemeyer, Alice C.

AU - Praeger, Cheryl E.

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N2 - Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the k-element subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n − 1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.

AB - Let H be a permutation group on a set Λ, which is permutationally isomorphic to a finite alternating or symmetric group An or Sn acting on the k-element subsets of points from {1, . . . , n}, for some arbitrary but fixed k. Suppose moreover that no isomorphism with this action is known. We show that key elements of H needed to construct such an isomorphism ϕ, such as those whose image under ϕ is an n-cycle or (n − 1)-cycle, can be recognised with high probability by the lengths of just four of their cycles in Λ.

KW - Symmetric and alternating groups in subset actions

KW - Large base permutation groups

KW - Finding long cycles

U2 - 10.13069/jacodesmath.28239

DO - 10.13069/jacodesmath.28239

M3 - Article

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VL - 2

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EP - 149

JO - Journal of Algebra Combinatorics Discrete Structures and Applications

JF - Journal of Algebra Combinatorics Discrete Structures and Applications

IS - 2

ER -

Identifying long cycles in finite alternating and symmetric groups acting on subsets

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EP/C523229/1: Multidisciplinary Critical Mass in Computational Algebra and Applications

Cite this