Discontinuous automorphisms of the proper Galilei and Euclidean groups (Q753969)

Let \(E^+_ n\) be the Euclidean group in n dimensions with no space inversions and let \(E_ n\) be the corresponding Galilei group. It is proved that the groups \(E^+_ 3\) and \(E_ 3\) have discontinuous automorphisms. All discontinuous automorphisms of \(E^+_ 3\) and of \(E_ 3\) arise from derivations of R. The cardinality of the set of all discontinuous automorphisms of \(E^+_ 3\) is \(2^{2^ N}\), where N is the cardinality of the set of all integers. All automorphisms of the group \(E^+_ n\) with \(n\geq 4\) are continuous.