I am trying to answer the following question:
Is there an injective group homomorphism between $\mathbb{Z}_{20}^{*}$ and $\mathbb{Z}_{64}^{*}$?
I have tried looking at the Euler function to see if the order of the image of such a bijective homomorphism, which is a subgroup of $\mathbb{Z}_{64}^{*}$, would divide $|\mathbb{Z}_{64}^{*}|$ (as would be expected by Lagrange). If this would not be true, then we have reached a contradiction, and there is no such homomorphism. I checked this indeed, and, as $8 \mid 32$, there is no contradiction.
I then thought perhaps the identity function could work - but I'm finding myself testing whether for each pair $\{a,b\}$ taken from the $8$ values in $\mathbb{Z}_{20}^{*}$ the following holds:
$$\phi(ab \pmod{20}) = \phi(a)\phi(b) \pmod{64}$$
which doesn't seem like the right direction (I'm going number-by-number here), but I'm short of other options - any ideas?
Also, generally speaking, is there a set of 'tools' that may be useful to keep in mind when trying to understand whether homomorphisms exist? Or how to find them / count them etc.?
And advice would be greatly appreciated.