TITLE: Does an abelian subgroup inject into the abelianisation of the whole group?
QUESTION [12 upvotes]: If $H <G $ are groups and H is abelian, do we get an injection from H into $G/[G,G] $?

REPLY [11 votes]: In fact, this happens all the time for a nonabelian group $G$. If $G$ is not abelian, then its commutator subgroup $[G,G]$ is nontrivial. Take any $x\in[G,G]$ different from $1$, and let $H$ be the subgroup generated by $x$. Then $H$ is abelian but maps to $0$ in the quotient $G/[G,G]$!