let be the dilation operator $ x \frac{d}{dx} $ i know this operator is invariant under the change $ y=ax$ for any positive 'a' real number
however let be the change $ y= \frac{-1}{x}$ then the operator $x \frac{d}{dx}= y \frac{d}{dy} $ still remains invariant .. is there a mathematical reason to see why this happens ?? , given the group of dilations in one dimension, is there a bigger group containing it ??