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Erschienen in:
Latin Squares Based on Groups Kapitel lesen Erstes Kapitel lesen

2018 | OriginalPaper | Buchkapitel

13. Groups of Small Order

verfasst von : Anthony B. Evans

Erschienen in: Orthogonal Latin Squares Based on Groups

Verlag: Springer International Publishing

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Abstract

In this chapter we will describe what is known about the structure of Orth(G) for G a group of small order. Much of this knowledge is the result of computer searches. We will summarize known values and bounds for |Orth(G)| and ω(G) when |G|≤ 23. We will see that order 23 marks a divide. We can find values of |Orth(G)| in the literature for every group of order 23 or less, but we can find no such value for any group of order 24, other than the cyclic group of order 25 and those that have cyclic Sylow 2-subgroups and hence do not admit orthomorphisms. It will be seen that we know ω(G) for all abelian groups G of order 19 or less but can only establish a lower bound for \(\omega (\mathbb {Z}_2\times \mathbb {Z}_2\times \mathbb {Z}_5)\); and we can only establish lower bounds for ω(G) for nonabelian groups G of order 16. We will also present what is known for groups of order greater than 23. Work on these larger groups has been concentrated on trying to improve lower bounds for ω(G) for groups G that are direct products of elementary abelian groups.

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Vorheriges Kapitel ω(G) for Some Classes of Nonabelian Groups
Nächstes Kapitel Projective Planes from Complete Sets of Orthomorphisms
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Metadaten
Titel
Groups of Small Order
verfasst von
Anthony B. Evans
Copyright-Jahr
2018
Buch
Orthogonal Latin Squares Based on Groups

Print ISBN: 978-3-319-94429-6

Electronic ISBN: 978-3-319-94430-2

Copyright-Jahr: 2018

DOI
https://doi.org/10.1007/978-3-319-94430-2_13