Let $G_k$ and $G_m$ be cyclic groups of orders $k,m$ respectively. Is there a way to count the number of epimorphisms from $G_m$ to (on) $G_k$?
Thank you.
Let $G_k$ and $G_m$ be cyclic groups of orders $k,m$ respectively. Is there a way to count the number of epimorphisms from $G_m$ to (on) $G_k$?
Thank you.
Take $G_n=\mathbb Z_n$. In order to have an epimorphism $f:\mathbb Z_m\to\mathbb Z_k$ we must have $k\mid m$ ($\overline 1\in\operatorname{Im}(f)$ $\Leftrightarrow$ $\exists\hat{a}\in\mathbb Z_m$ such that $f(\hat{a})=\overline 1$ $\Rightarrow$ $mf(\hat{a})=\overline m$ $\Rightarrow$ $\overline 0=\overline m$ $\Rightarrow$ $k\mid m$). Furthermore, every homomorphism will be determined by its value on $\hat 1$. Since we want the image to cover all $\mathbb Z_k$, then the image of $\hat 1$ must be invertible (as an element of the ring $\mathbb Z_k$). Conclusion: we have $\phi(k)$ epimorphisms, where $\phi$ is the Euler totient function.