I am working through Audrey Terras's "Fourier Analysis on Finite Groups and Applications" and am confused about one of the exercises in chapter 2. Towards the bottom of page 45, there is an exercise about the interaction between the DFT and the dilation operator:
Let $a$ be an element of the unit group $(\mathbb{Z}/n\mathbb{Z})^*$. Define the dilation operator $D_a f(x) = f(ax)$, for all $x \in \mathbb{Z}/n\mathbb{Z}$. Show that
$$\mathfrak{F}D_af(y) = D_{a^{-1}}\mathfrak{F}f(y),$$
where $\mathfrak{F}$ denotes the DFT.
I wrote out both sides in terms of the definitions of the dilation operator and the DFT, but am not sure what to do from there. I saw that for an analogous property in the continuous case you need to do a variable substitution. I thought that was an obvious approach here as well, but I didn't see how to get that to work.
Do I actually need to do a variable substitution? The simplicity of the problem suggests to me that I am misunderstanding something. I was able to solve the previous problem about the DFT and the translation operator, which was just a matter of writing out both sides, so it seems like this problem should be pretty straightforward as well.
If someone could point me in the right direction, I would greatly appreciate it!