The equivalence of two cyclic objects on \(pq\) elements
type: Item
instance of: scholarly article
title: The equivalence of two cyclic objects on \(pq\) elements
zbMATH Open document ID: 0859.05014
DOI: 10.1016/0012-365X(94)00343-H
author: W. Cary Huffman
published in: Discrete Mathematics
publication date: 1997-04-07T00:00:00Z
review text: A cyclic object on \(n\) elements is a combinatorial structure on \(\{0,1,\dots,n-1\}\) which has an \(n\)-cycle in its automorphism group. Examples of classes of cyclic objects are circulant graphs or digraphs or cyclic block designs. For any such class the symmetric group on \(\{0,1,\dots,n-1\}\) induces an equivalence relation. Since most permutations, when applied to a cyclic object, will produce an object that is not cyclic, the equivalence classes are much smaller than in the general case without cyclicity assumption. Depending on the object class, and on the number-theoretic properties of \(n\), it has been shown in many cases that only equivalence by multipliers is possible. Pálfy showed that this is the case for any class of cyclic objects if \(\text{gcd}(n,\varphi(n))=1\). In this paper, the author shows that for \(n=pq\) with \(p\), \(q\) primes in the case \(\text{gcd}(n,\varphi(n))\neq1\) there are still only few possibilities: there is a list of at most \(\varphi(n)\) permutations such that equivalence of cyclic objects on \(n\) elements is possible only by these permutations. They have a piecewise structure of a multiplier.
Mathematics Subject Classification ID: 05A99
Mathematics Subject Classification ID: 20B25
Mathematics Subject Classification ID: 05E20
Mathematics Subject Classification ID: 05C60
zbMATH DE Number: 906895
zbMATH Keywords: equivalence testing
zbMATH Keywords: isomorphism testing
zbMATH Keywords: cyclic object
zbMATH Keywords: automorphism group
zbMATH Keywords: circulant graphs
zbMATH Keywords: digraphs
zbMATH Keywords: cyclic block designs
zbMATH Keywords: symmetric group
zbMATH Keywords: permutations
zbMATH Keywords: equivalence by multipliers
reviewed by: Peter Brass
MaRDI profile type: Publication
cites work: Isomorphism of circulant graphs and digraphs
cites work: Isomorphism problem for a class of point-symmetric structures
cites work: Topological invariants of 2-designs arising from difference families
cites work: On Cyclic Steiner 2-Designs
cites work: Multipliers and generalized multipliers of cyclic objects and cyclic codes
cites work: Duadic Codes
cites work: Q4146667
cites work: Isomorphism problem for relational structures with a cyclic automorphism
cites work: Q3791159
cites work: Q5752581
cites work: Isomorphism of strong starters in cyclic groups
cites work: Q3996427
full work available at URL: 0012-365x(94)00343-h
OpenAlex ID: W2137999379
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description: scientific article; zbMATH DE number 906895