### 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 *Φ _{i}* :

*Y*→ G for

_{i}*i*= 1, 2, 3 such that

*Φ*

_{1}(

*y*

_{1})

*Φ*

_{2}(

*y*

_{2}) =

*Φ*

_{3}(

*y*

_{3}) for each (

*y*

_{1},

*y*

_{2},

*y*

_{3}) ∈

*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,b

^{a}]〉. Our proof uses computations to answer questions about finitely presented groups which are known to be algorithmically undecidable in general.

Original language | English |
---|---|

Pages (from-to) | 142-152 |

Number of pages | 11 |

Journal | Journal of Symbolic Computation |

Volume | 86 |

DOIs | |

Publication status | Published - 2018 |

### Keywords

- Baumslag group
- Finitely presented group
- Group embedding
- Partial Latin square