TITLE: Group is indecomposible but homomorphic image is not
QUESTION [5 upvotes]: 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.

REPLY [9 votes]: Consider the quotient map $\Bbb Z \to \Bbb Z/6 \cong \Bbb Z/2 \oplus \Bbb Z/3$