Transformations that Preserve Normality of Distribution

Question

Ok, so Let X be normally-distributed. For simplicity, assume X $ \~ N(0,1)$

We know Affine transformations preserve the normality of X, i.e., if X is

normal, so are $aX$ and $aX +b$ for $a,b$ Real. I guess we can show this

using either the MGF or just the general form $\frac {1}{\sqrt 2\pi} e^{\frac {(x-\mu)^2}{\sigma}}$

And pretty sure the transformations that preserve normality are a group.

Are there transformations other than the affine ones that preserve the

normality of $X$

Answer 1

What about $\varphi(X)$ where $\varphi(X)=-X \text{ when }|X|<1$ and $\varphi(X)=X$ otherwise?