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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.

Shaun
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Anon
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    Any number coprime to $20$ must be odd, and so it must also be coprime to $64$ (since $2$ is the only prime number dividing $64$). – Geoffrey Trang Dec 31 '21 at 14:45
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    Helpful links: https://math.stackexchange.com/questions/314846/for-what-n-is-u-n-cyclic, https://math.stackexchange.com/questions/1387378/show-that-u-mathbb-z-pn-is-cyclic-by-considering-the-order-of-1p , https://en.wikipedia.org/wiki/Multiplicative_group_of_integers_modulo_n . Those let you characterise $\mathbb Z_n^$ for any $n\in\mathbb N$. You will end up with $\mathbb Z_{20}^\cong C_2\times C_4$ and $\mathbb Z_{64}^*\cong C_2\times C_{16}$ (if I have not made some mistake). –  Dec 31 '21 at 14:53
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    Thus, there will be an injective homomorphism $C_2\times C_4\to C_2\times C_{16}$ given by $(a,b)\mapsto (a', b'^4)$ where $a,b,a',b'$ are the generators in $C_2, C_4, C_2$ (2nd copy), $C_{16}$ respectively. –  Dec 31 '21 at 14:54
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    @StinkingBishop slight nitpicking: generators are not unique in general, so speaking of "the generators" is slightly off imo. – Peter Müller Dec 31 '21 at 15:01
  • @PeterMüller You are right, I should've been more precise. However, my biggest concern here is whether the whole advice (the links, warts and all) has been useful to the OP or not. –  Dec 31 '21 at 15:57

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