arXiv:1507.00638v1 [math.GR] 2 Jul 2015 The multiplication groups of topological loops ´ Agota Figula Abstract In this short survey we discuss the question which Lie groups can occur as the multiplication groups Mult(L) of connected topological loops L and we describe the correspondences between the structure of the group Mult(L) and the structure of the loop L. 2010 Mathematics Subject Classification: 57S20, 22E25, 20N05, 57M60 Key words and phrases: multiplication group of loops, topological transformation group, solvable Lie groups, filiform Lie groups 1. Introduction A loop (L, ·) is a quasigroup with identity element e ∈ L. The left translations λ a : y → a · y and the right translations ρ a : y → y · a, a ∈ L, are bijections of L. To obtain closer relations between a loop and a group one has to investigate the groups which are generated by the translations of a loop L. The group generated by all left and right translations of L is called the multiplication group Mult(L) of L (cf. [1], [2]). The subgroup G of Mult(L) generated by all left translations of L is the group of left translations of L. The group Mult(L) reflects well the normal structure of the corresponding loop L, since for any normal subloop of L there is a normal subgroup in Mult(L) and for every normal subgroup N of Mult(L) the orbit N (e) is a normal subloop of L. Hence, it is an interesting question which groups can be represented as multiplication groups of loops. The criterion for the decision whether a group is the multiplication group of a loop L is given in [16]. This criterion has been successfully applied in particular in the case of Lie groups. In [15] topological and differentiable loops L having a Lie group as the group G of left translations of L are studied. There the topological loops L are treated as continuous sharply transitive sections σ : G/H → G, where H is the stabilizer of e ∈ L in G. Publications, in which classes of connected topological loops L have been classified (cf. [15], [5]) show that there are only few Lie groups which are not groups of left translations of a topological loop. If L is a topological loop having a Lie group as the group of left translations, then the group Mult(L) is prevalently a differentiable transformation group of infinite dimension. The condition for Mult(L) to be a Lie group is a 1