I was recently asked this in Abstract Algebra class on group theory:     Let G be a group and G' its commutator subgroup (i.e. its minimal subgroup containing all commutators $ [x,y] = xyx^{-1}y^{-1} $ where $ x,y \in G $ ). Where are asked to show G' is the minimal normal subgroup such that the quotient group $  G/G' $ is Abelian (if $ H \unlhd G $ and G/H is Abelian then we must have $ G' &lt; H $)   To be honest I have tried but could not seem to be able to solve this. I have shown G' is normal and quotient group $ G/G' $ is abelian, but I cannot prove G' is minimal normal subgroup with this property. I really would appreciate the help on this. Thanks all
Good day. I'm not too sure how I should go about asking my question, so if it is in anyway confusing, feel free to edit it.   In my Group Theory course we defined and showed a few properties of the commutator subgroup $[G,G]$ of a group $G$. And then we stated and proved a lemma in lecture that I'm having some difficulty wrapping my head around. I'll state it here for clarity.  Lemma a) The commutator subgroup $[G,G]$ of $G$ is a normal subgroup of $G$. b) The quotient group $G / [G,G]$ is the largest abelian quotient of G in the sense that every homomorphism $\varphi : G \to A$ where $A$ is an an abelian group, induces a unique homomorphism $\bar \varphi : G/[G,G] \to A$ so that $\varphi =\bar \varphi \circ \pi$ where $\pi : G \to G/[G,G]$ is the quotient homomorphism.  Now, it's not the proof of this lemma that I'm having trouble with, it's the italicized text that is messing with me. I completely understand what it is that the italicized text is saying, I'm just not too sure why it is that the property gives us the "largeness" that is claimed. I can see that the above forces $G$ to "pass through" $G/[G,G]$ before it "reaches" $A$ (I apologize for the casual terminology), so I can sort of see why $G/[G,G]$ should be the largest quotient, in a non-rigorous sort of way.  Overall I'm very confused. Perhaps I'm missing something very simple. Any help or nudges in the right direction would be much appreciated.   As a side note I did try a search on Google and here on Stack Exchange and couldn't find anything that answered my question. So if I missed something obvious, then my apologies. Thank you in advance.
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