Let $f:G\to H$ be a homomorphism of groups.
Is it true that $G/[G,G]\cong f(G)/[f(G),f(G)]$, where $[,]$ denotes the commutator subgroup?
2026-04-23 14:26:29.1776954389
Properties of the commutator subgroup
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No, let $G=\Bbb Z$ and $H=D_8$ with $f(x)=e$ Then we have $$\Bbb Z/[\Bbb Z,\Bbb Z]=\Bbb Z/e=\Bbb Z$$ but $$f(\Bbb Z)=e$$ and as such $$e/[e,e]=e$$ Which are clearly not isomorphic.