# If $g:\mathbb{Z_{10}}\rightarrow U_{20}$ is a group homomorphism, then the order of $g(1)$ is either $1$ or $2$. [closed]

Why is that if $$g:\mathbb{Z_{10}}\rightarrowU_{20}$$ is a group homomorphism, then the order of $$g(1)$$ is either $$1$$ or $$2$$?

Also, $$g$$ is a function, $$\mathbb{Z_{10}}$$ is the group of integers modulo $$10$$, and $$U_{20}$$ is the group of units modulo $$20$$.

## closed as off-topic by Shaun, Gibbs, José Carlos Santos, John B, BrahadeeshDec 10 '18 at 12:07

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• $1$ has order $10$ in $\mathbb{Z}_{10}$. – Randall Dec 10 '18 at 5:18

Hint: $$\operatorname{Im}(g)$$ is a subgroup of $$U_{20}$$ and $$1$$ (as an element of $$\Bbb Z_{10}$$) generates $$\Bbb Z_{10}$$.
In fact, $$U(20)=\mathbb Z_4\oplus \mathbb Z_2$$. So $$\operatorname{Im}g=\langle g(1)\rangle$$ has order dividing $$10$$ and $$4$$.