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

Abstract

Suppose that Y1, Y2, Y3 are finite sets and P⊆Y1×Y2×Y3. We say that P embeds in a group G if there exist injective maps ϕi:Yi→G for i=1,2,3 such that ϕ1(y1)ϕ2(y2)=ϕ3(y3) for each (y1,y2,y3)∈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=〈a,b|b=[b,ba]〉. Our proof uses computations to answer questions about finitely presented groups which are known to be algorithmically undecidable in general.