Small partial Latin squares that embed in an infinite group but not into any finite group. (May 2018)

• Record Type: Journal Article
• Authors: Dietrich, Heiko; Wanless, Ian M.
• Is Part Of: Journal of symbolic computation. Volume 86(2018)
• Page Start: 142
• Page End: 152
• Publication Date: 2018-05
• Subjects: Partial Latin square -- Group embedding -- Finitely presented group -- Baumslag group; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2017.04.002
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: Suppose that Y 1, Y 2, Y 3 are finite sets and P ⊆ Y 1 × Y 2 × Y 3 . We say that P embeds in a group G if there exist injective maps ϕ i : Y i → G for 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.