# The spectrum of group-based Latin squares

## TL;DR

This paper explores the existence of group-based Latin squares, providing constructions for certain groups and characterizing when such Latin squares exist based on group properties.

## Contribution

It introduces new constructions for Latin squares based on semi-direct product groups and characterizes their existence for various group orders.

## Key findings

- Group-based Latin squares exist for orders 1, 2, and 4.
- Existence of non-abelian groups determines Latin square existence for other orders.
- Constructed Latin squares for many semi-direct product groups.

## 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 $n$.