2. (i) Let be the totality of points x = (XI" . . , x,,) for which all coordinates are different from zero, and let G be the group of transformations x; = cx i, e > O. Then a maximal invariant under G is (sgn x"' XI/X" , . . . , X,, _I/X,,) where sgn X is 1 or -1 as x is positive or negative. (ii) Let be the space of points x = (XI" .. , x,,) for which all coordinates are distinct, and let G be the group of all transformations x; = !(x;), i = 1, . .. , n, such that! is a 1: 1 transformation of the real line onto itself with at most a finite number of discontinuities. Then G is transitive over [(ii): Let x = (XI"' " x,,) and x' = (x], ... , x~) be any two points of Let 11" . . ,1" be a set of mutually exclusive open intervals which (together with their end points) cover the real line and such that xj E lj' Let 'i, ..., l~ be a corresponding set of intervals for x;,..., x~ Then there exists a transformation ! which maps each lj continuously onto ';, maps xj into xj, and maps the set of n - 1 end points of ll"'" '" onto the set of end points of 'r, .. .,l,; .]