Suppose $K$ is a non finitely generated and residually finite group, is it possible that $K$ has finite abelianization, i.e. the quotient group $K \big/ [K,K]$ is finite?
If we take $K$ to be the infinitely generated free group, then the abelianization (which is the free abelian group) would be infinite.
On the other hand, if we take $K$ to be an infinitely generated abelian group, then the abelianization would clearly be infinite.
Then I thought that maybe $K$ can be a semidirect product of two groups? I have not seen many examples of infinitely generated semidirect product groups. Any help would be really appreciated.
Take your favourite perfect group $G\neq1$ and the direct product $H = \prod\limits_{n=1}^\infty G_n$, where each $G_n=G$. The commutators of $H$ are taken coordinate-wise, so now $[H, H] = \prod\limits_{n=1}^\infty [G_n, G_n] = \prod\limits_{n=1}^\infty G_n$, and so $H/[H, H]=1$ but $H$ itself is not finitely generated.
Edit (YCor) as mentioned in the comments, the above-claimed equality $[H,H]=\prod [G_n,G_n]$ can fail in general. It holds if (and only if) there exists $m$ such that every element of $[G,G]$ is a product of $\le m$ commutators. This trivially holds if $G$ is finite.