Abstract
We construct sequencings for many groups that are a semi-direct product of an odd-order abelian group and a cyclic group of odd prime order. It follows from these constructions that there is a group-based complete Latin square of order $n$ if and only if $n \in \{ 1,2,4\}$ or there is a non-abelian group of order $n$.
Highlights
A Latin square of order n is an n×n array of symbols from a set of size n with each symbol appearing once in each row and once in each column
A Latin square is row-complete or Roman if each pair of distinct symbols appears in adjacent positions in a row once in each order
Interest in complete and row-complete Latin squares was originally prompted by their usefulness in the design of experiments where neighboring treatments, whether in space or time, might interact
Summary
A Latin square of order n is an n×n array of symbols from a set of size n with each symbol appearing once in each row and once in each column. The principal question for this work is to determine at what orders there is a group for which it is possible to permute the rows and columns of its Cayley table to give a complete Latin square. [11] The rows and columns of the Cayley table of a group G of order n may be permuted to give a complete Latin square if and only if G is sequenceable. We construct sequencings for some semi-direct products Zq A where A is an abelian group of odd order and q is an odd prime, including all possible such groups when A is cyclic These constructions allow us to determine the full spectrum of orders at which a group-based complete Latin square exists: Theorem 2.
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