Classification of highly symmetrical translation loops of order 2 p, p prime

Mikhail Klin; Nimrod Kriger; Andrew Woldar

doi:10.1007/s13366-012-0132-4

Classification of highly symmetrical translation loops of order 2 p, p prime

Mikhail Klin, Nimrod Kriger, Andrew Woldar

Department of Mathematics

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

2 Scopus citations

Abstract

Using tools from algebraic graph theory, we prove that for every odd prime p there is just one proper loop of order 2p such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order 4p². Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.

Original language	English
Pages (from-to)	253-276
Number of pages	24
Journal	Beitrage zur Algebra und Geometrie
Volume	55
Issue number	1
DOIs	https://doi.org/10.1007/s13366-012-0132-4
State	Published - 1 Mar 2014

Keywords

3-net
4-vertex condition
Association scheme
Cayley graph
Finite permutation group
G-loop
Highly symmetrical loop
Intercalate
Loop
Partial difference set
Primitive S-ring
Strongly regular graph
Translation loop
Transversal design

ASJC Scopus subject areas

Algebra and Number Theory
Geometry and Topology

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10.1007/s13366-012-0132-4

Cite this

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title = "Classification of highly symmetrical translation loops of order 2 p, p prime",

abstract = "Using tools from algebraic graph theory, we prove that for every odd prime p there is just one proper loop of order 2p such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order 4p2. Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.",

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AU - Klin, Mikhail

AU - Kriger, Nimrod

AU - Woldar, Andrew

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PY - 2014/3/1

Y1 - 2014/3/1

N2 - Using tools from algebraic graph theory, we prove that for every odd prime p there is just one proper loop of order 2p such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order 4p2. Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.

AB - Using tools from algebraic graph theory, we prove that for every odd prime p there is just one proper loop of order 2p such that the automorphism group of the corresponding rank 5 association scheme contains a regular subgroup of order 4p2. Although our results were initially obtained on the basis of theoretical generalization of a tremendous number of computer algebra experiments, our final computer-free proof uses quite elementary arguments from group theory and combinatorics. In this paper we provide this computer-free proof, as well as discuss further suggested use of our methods.

KW - 3-net

KW - 4-vertex condition

KW - Association scheme

KW - Cayley graph

KW - Finite permutation group

KW - G-loop

KW - Highly symmetrical loop

KW - Intercalate

KW - Loop

KW - Partial difference set

KW - Primitive S-ring

KW - Strongly regular graph

KW - Translation loop

KW - Transversal design

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Classification of highly symmetrical translation loops of order 2 p, p prime

Abstract

Keywords

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