I am currently studying about exact sequence. To illustrate that not all short exact sequences are split, my teacher provided an example in the homework.
For every $m \geqslant2$ construct a short exact sequence of abelian groups
$$0\rightarrow \mathbb{Z}_m \rightarrow \mathbb{Z}_{m^2}\rightarrow \mathbb{Z}_m\rightarrow 0.$$
At first I tried the following:
Define the homomorphism from $\mathbb{Z} _2$ to $\mathbb{Z}_{m^2}$ as identity map $id_{\mathbb{Z}_m}$. The homomorphism $\pi$ from $\mathbb{Z}_{m^2}$ to $\mathbb{Z}_2$ as following:
$$\pi(\bar{n})=\bar{0} , 0\leqslant n\leqslant m-1,\pi(\bar{n})=n\quad mod\quad m, m\leqslant n\leqslant m^2-1.$$
Then we have $Im(id_{\mathbb{Z_m}})=Ker(\pi)$. But I quickly realized $\pi$ is NOT a group homomorphism.
Can anyone give me some hints? Any idea will be appreciated! Thanks in advance!