Let $G_i$ be abelian Groups. A exact sequence of the form $ 0 \to G_1 \to G_2 \to G_3 \to 0$ is called a short exact sequence. Is the following statement true?
If $G_3$ is finitely generated then there is a short exact sequence with $G_2$ and $G_1$ free groups?
I'm unable in producing a counter example.please help.
I assume you mean that a sequence $$0\to G_1 \to G_2\to G_3\to 0\tag{*}$$is short exact. Set $$G_3=\Bbb Z_{a_1}\times \Bbb Z_{a_2}\times \cdots \Bbb Z_{a_n}\times \Bbb Z^m$$ If $G_1=\Bbb Z^n, G_2=\Bbb Z^{n+m}$ and $f:G_1\to G_2$ is given by first multiplying each component with the corresponding $a_i$, then adding $m$ zeroes to the end, the sequence $(*)$ is exact.