# Small partial Latin squares that embed in an infinite group but not into any finite group

## TL;DR

This paper determines the minimal size of small partial Latin squares that can embed in an infinite group but not in any finite group, providing explicit examples and leveraging computational methods.

## Contribution

It proves the minimal size is 12, identifies 50 such examples up to equivalence, and shows they embed in the Baumslag group, advancing understanding of embeddings in group theory.

## Key findings

- Minimal size of such partial Latin squares is 12
- 50 examples of size 12 exist up to equivalence
- All examples embed in the Baumslag group

## Abstract

Suppose that $Y_1,Y_2,Y_3$ are finite sets and $P\subseteq Y_1\times Y_2\times Y_3$. We say that $P$ embeds in a group $G$ if there exist injective maps $\phi_i\colon Y_i\rightarrow G$ for $i=1,2,3$ such that $\phi_1(y_1)\phi_2(y_2)=\phi_3(y_3)$ for each $(y_1,y_2,y_3)\in P$. Hirsch and Jackson asked for the cardinality of the smallest $P$ that embeds in some infinite group but not into any finite group. We prove that the answer to their question is 12. Moreover, we show that there are 50 examples of cardinality 12, up to equivalence, and each of them embeds in the (infinite) Baumslag group $G=\langle a,b \mid b=[b,b^a]\rangle$. Our proof uses computations to answer questions about finitely presented groups which are known to be algorithmically undecidable in general.