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.

We determine the 6-dimensional nilpotent metric Lie algebras such that the Lie algebra $${\mathfrak {n}}$$ n has a descending series of ideals invariant under all automorphisms of $${\mathfrak {n}}$$ n and the dimension of the consecutive members of the series decreases by one. We call them metric Lie algebras having a framing determined by ideals. We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on 6-dimensional nilpotent Lie groups having a Lie algebra $${\mathfrak {n}}$$ n as their Lie algebra.

We determine the 6-dimensional nilpotent metric Lie algebras such that the Lie algebra n has a descending series of ideals invariant under all automorphisms of n and the dimension of the consecutive members of the series decreases by one. We call them metric Lie algebras having a framing determined by ideals. We classify the isometry equivalence classes and determine the isometry groups of connected and simply connected Riemannian nilmanifolds on 6-dimensional nilpotent Lie groups having a Lie algebra n as their Lie algebra.Mathematics Subject Classification (2010). Primary 53C30; Secondary 57M60, 17B30, 22E25.

The aim of our study was to explore the effects of the active use of realistic exercises in the field of trigonometry. We taught a group of 14 pupils, who were in grade 11. The most of them told us they did not plan mathematics-related studies in the future. We included realistic exercises into our teaching plan, which covered the fields of scalar product, as well as the sine and cosine theorems. Our teaching experiment was done within the framework of online teaching. Effects on the motivation, performance and results of the students were taken into consideration. We also attempted to examine the effects of online teaching on motivation and whether the use of realistic exercises is worthwhile in an online classroom environment. Performance of the students showed a tendency of improvement when they were dealing with the material through realistic exercises even despite the teaching happened online. Subject Classification: 97C70, 97D40, 97G60