# Existence of generalized Latin squares which are not embeddable in any group

## Abstract

Let *n* be a positive integer. A generalized Latin square of order *n* is an \(n\times n\) matrix such that the elements in each row and each column are distinct. In this paper, we show that for any integer \(n\ge 6\) and any integer *m* where \(m\in \left\{ n, n+1, \dots , \frac{n(n+1)}{2}-2\right\} \), there exists a commutative generalized Latin square of order *n* with *m* distinct elements which is not embeddable in any group. In addition, we show that for any integer \(r\ge 3\) and any integer *s* where \(s\in \{ r, r+1, \dots , r^2-2\}\), there exists a non-commutative generalized Latin square of order *r* with *s* distinct elements which is not embeddable in any group.

## Keywords

Generalized Latin square Embeddable in a group Partial Latin square## Mathematics Subject Classification

05B15 20D60## Notes

### Acknowledgements

The authors would like to thank the referee for many helpful comments which led to an improved paper. The corresponding author is grateful for financial support given by the University of Malaya Research Grant UMRG337/15AFR.

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