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RESOLVABLE MENDELSOHN DESIGNS AND FINITE FROBENIUS GROUPS

Part of: Designs and configurations

Published online by Cambridge University Press:  30 May 2018

D. F. HSU  and
SANMING ZHOU Open the ORCID record for SANMING ZHOU [Opens in a new window]
D. F. HSU
Affiliation:
Department of Computer and Information Sciences, Fordham University, New York, NY 10023, USA email hsu@cis.fordham.edu
SANMING ZHOU*
Affiliation:
School of Mathematics and Statistics, The University of Melbourne, Parkville, VIC 3010, Australia email sanming@unimelb.edu.au
*
✉sanming@unimelb.edu.au
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Abstract

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We prove the existence and give constructions of a $(p(k)-1)$-fold perfect resolvable $(v,k,1)$-Mendelsohn design for any integers $v>k\geq 2$ with $v\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}\,k$ such that there exists a finite Frobenius group whose kernel $K$ has order $v$ and whose complement contains an element $\unicode[STIX]{x1D719}$ of order $k$, where $p(k)$ is the least prime factor of $k$. Such a design admits $K\rtimes \langle \unicode[STIX]{x1D719}\rangle$ as a group of automorphisms and is perfect when $k$ is a prime. As an application we prove that for any integer $v=p_{1}^{e_{1}}\cdots p_{t}^{e_{t}}\geq 3$ in prime factorisation and any prime $k$ dividing $p_{i}^{e_{i}}-1$ for $1\leq i\leq t$, there exists a resolvable perfect $(v,k,1)$-Mendelsohn design that admits a Frobenius group as a group of automorphisms. We also prove that, if $k$ is even and divides $p_{i}-1$ for $1\leq i\leq t$, then there are at least $\unicode[STIX]{x1D711}(k)^{t}$ resolvable $(v,k,1)$-Mendelsohn designs that admit a Frobenius group as a group of automorphisms, where $\unicode[STIX]{x1D711}$ is Euler’s totient function.

Keywords

Mendelsohn designbalanced directed cycle designresolvable Mendelsohn designperfect Mendelsohn designFrobenius groupfixed-point-free automorphism

MSC classification

Primary: 05B05: Block designs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 98 , Issue 1 , August 2018 , pp. 1 - 13
Copyright
© 2018 Australian Mathematical Publishing Association Inc. 

Footnotes

The second author was supported by a Future Fellowship (FT110100629) of the Australian Research Council.

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