How do the images $\pi(g)$ of the canonical projection $\pi: G\to G/[a,b]$ look like?

Question

I am currently working on a proof that uses the canonical projection $$\pi: G \to G/[G,G].$$

$G/[G,G]$ is the quotientgroup and its elements are cosets of $[G,G]$. Thus i was assuming that for any $g \in G$ the image $\operatorname{Im} \pi$ is simply given by $$\operatorname{Im} \pi = \{g[G,G]\mid g \in G\}$$

Therefore we have $$\pi(g) = g[G,G]$$

Is that correct?

Thanks for any help!

Answer 1

The image is just the group of equivalence classes. For any element $g \in G$, $\pi(g) = [g]_{[G,G]}$. Where $[g]_{[G,G]}$ is the equivalence class of elements that are congruent to g modulo $[G,G]$. We can also write this equivalence class as $g + [G,G]$.