I'm looking to find a homomorphism from trivial group to $\mathbb{Z}_{5n}$. Is it simple as $$g: e \to \mathbb{Z}_{5n}, g(x) = 5n$$
Because $$g(1) = g(1 \cdot 1) = 5n = 0 (\text{mod 5n})$$ and $$g(1) \oplus g(1) = 5n + 5n = 0 (\text{mod 5n})$$
And the identity is preserved with this homomorphism. Is this correct or am I missing something?