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Published in:
2018 | OriginalPaper | Chapter
11. Extensions of Orthomorphism Graphs
Author : Anthony B. Evans
Published in: Orthogonal Latin Squares Based on Groups
Publisher: Springer International Publishing
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Abstract
In previous chapters we considered orthomorphisms and orthomorphism graphs of elementary abelian groups. Other than these groups, the other classes of abelian groups that have received significant attention are the cyclic groups and direct products of elementary abelian groups. In this chapter we will define the extension of the orthomorphism graph of a group G by a group H: this is an orthomorphism graph of G × H. We will discuss two special cases: H = GF(3)+ and G = GF(q)+ and \(H=\mathbb {Z}_3\) and \(G=\mathbb {Z}_m\), \(\gcd (m,3)=1\). We will use the 1978 construction of a set of four MOLS of order 15 by Schellenberg, Van Rees, and Vanstone to introduce extensions of orthomorphism graphs. We will discuss extensions by GF(3)+, and we will use difference equations to study the extension of Orth(GF(q)+) by GF(3)+. We will consider the special case of extensions of the orthomorphism graph of GF(q)+ consisting of linear orthomorphisms of GF(q)+. We will extend these methods to the study of extensions of cyclic groups by GF(3)+.