A nontrivial homomorphic image of an indecomposable group need not be indecomposable.
A group $G$ is indecomposable if $G \neq \{ e\}$ and G is not the (internal) direct product of two of its proper subgroups.
Let $f$ be a homomorphism such that it is non trivial an $G$ be an indecomposible group. I have to find a homomorphism after finding a indecomposible group. Every Simple group is indecomposible, $\mathbb{Z}$ , $\mathbb{Z}_{p^n}$ and $S_n$ are indecomposible.
But I am unable to find a example of homomorphism to prove that $f(P)$ is not indecomposible.
Best Answer
Consider the quotient map $\Bbb Z \to \Bbb Z/6 \cong \Bbb Z/2 \oplus \Bbb Z/3$