Ameer Mohammed Hussein Hasan Al-abayechi

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.

The main result of our consideration is the proof of the centrally nilpotency of class two property for connected topological proper loops of dimension which have an at most six-dimensional solvable indecomposable Lie group as their multiplication group. This theorem is obtained from our previous classification by the investigation of six-dimensional indecomposable solvable multiplication Lie groups having a five-dimensional nilradical. We determine the Lie algebras of these multiplication groups and the subalgebras of the corresponding inner mapping groups.

Using connected transversals we determine the six-dimensional indecomposable solvable Lie groups with five-dimensional nilradical and their subgroups which are the multiplication groups and the inner mapping groups of three-dimensional connected simply connected topological loops. Together with this result we obtain that every six-dimensional indecomposable solvable Lie group which is the multiplication group of a three-dimensional topological loop has one-dimensional centre and two- or three-dimensional commutator subgroup.

In this paper we deal with the class of decomposable solvable Lie groups having dimension six. We determine those Lie groups in and their subgroups which are the multiplication groups Mult(L) and the inner mapping groups Inn(L) for three-dimensional connected simply connected topological loops L. This result completes the classification of the at most 6-dimensional solvable multiplication Lie groups of the loops L. Moreover, we obtain that every at most 3-dimensional connected topological proper loop having a solvable Lie group of dimension at most six as its multiplication group is centrally nilpotent of class two.