Dimension of homology of groups

Question

I am troubles when understanding why the following holds: Let $G$ be a group and $p\colon G \mapsto H$ a surjective homomorphism. Then, if $H_{1}(G, \mathbb{Q})$ is finite dimensional, so is $H_{1}(H,\mathbb{Q})$.

Can someone tell me from where does this derive? Thanks in advance!

Answer 1

$H_1(G,\mathbb{Z})=G/[G,G]$, thus if $f$ is surjective it induces a surjection

$G/[G,G]\rightarrow H/[H,H]=H_1(H,\mathbb{Z})$.

Use the fact that $H_1(G,\mathbb{Q})=H_1(\mathbb{Z})\otimes \mathbb{Q}$.