Is $(\mathbb{Z_4},+) \rightarrow (\mathbb{Z_5^{*}},\cdot), n\bmod 4 \mapsto n \bmod 5 $ a homomorphism?

Question

For the following relation

$(\mathbb{Z_4},+) \rightarrow (\mathbb{Z_5^{*}},\cdot), n\bmod 4 \mapsto n \bmod 5 $

Determine if it is well-defined and an homomorphism

So I think it is not well-defined because $6\equiv 2 \text{ ( mod 4)}$, but $6\not \equiv 2 \text{ ( mod 5)}$, then it can't be a homomorphism either.

What do you think?

Answer 1

You're correct. Nevertheless, there exist isomorphisms between these groups, bu they're defined through generators of the group $\mathbf Z^\times$, which are $2$ and $3$, say via $2$: \begin{align} \mathbf Z/4\mathbf Z&\longrightarrow(\mathbf Z/5\mathbf Z)^\times\\ k\bmod4&\longmapsto 2^{k\bmod 4}\bmod 5 \end{align}