We prove that the solvability of the multiplication group Mult(L) of a connected simply connected topological loop L of dimension three forces that L is classically solvable. Moreover, L is congruence solvable if and only if either L has a non-discrete centre or L is an abelian extension of a normal subgroup ℝ by the 2-dimensional nonabelian Lie group or by an elementary filiform loop. We determine the structure of indecomposable solvable Lie groups which are multiplication groups of three-dimensional topological loops. We find that among the six-dimensional indecomposable solvable Lie groups having a four-dimensional nilradical there are two one-parameter families and a single Lie group which consist of the multiplication groups of the loops L. We prove that the corresponding loops are centrally nilpotent of class 2.
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The paper is supported by the National Research, Development and Innovation Office (NKFIH) Grant No. K132951 and by the EFOP-3.6.1-16-2016-00022 project. The latter project is co-financed by the European Union and the European Social Fund.
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FIGULA, Á., AL-ABAYECHI, A. TOPOLOGICAL LOOPS HAVING SOLVABLE INDECOMPOSABLE LIE GROUPS AS THEIR MULTIPLICATION GROUPS. Transformation Groups 26, 279–303 (2021). https://doi.org/10.1007/s00031-020-09604-1