I don't get the point when I see the theorem in the below figure. What does $G'\leq\ker\varphi$ want to tell (I know the mathematical meaning word by word, but don't know the intuitive meaning)? What does "factor through" mean? Only if I feel natural with it, then I can remember it.
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Eric
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The Homomorphism Lemma states the following. Let $f\colon G\rightarrow H$ be a group homomorphism and $N$ be a normal subgroup of $G$ with $$ N\subseteq \ker(f). $$ Then there exists a unique group homomorphism $g:G/N\rightarrow H$ such that $f=g\circ \pi$, where $\pi:G\rightarrow G/N$ is the canonical surjective homomorphism. Hence $f$ factorizes as $g\circ \pi$. We need the condition $N\subseteq \ker(f)$ in oder that $g$ is well-defined.
Dietrich Burde
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Dietrich Burde, would you say that $f$ factors through $N$ or through $G/N$? (This worried me about the statement in the OP's book). – ancient mathematician Apr 25 '19 at 15:21
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1@ancientmathematician Through $G/N$, yes. See this post. – Dietrich Burde Apr 25 '19 at 15:24
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My view, too. I think "buy a better book" is in order! – ancient mathematician Apr 25 '19 at 15:26
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It's Dummit & Foote's Abstract Algebra. I'd heard many people complain about it, such as the organization or presentation. I rarely read this book. – Eric Apr 25 '19 at 15:29
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I see! Thank you all! PS: I have never heard about homomorphism lemma before. And having known the meaning of factor through, I feel more natural about this theorem. – Eric Apr 25 '19 at 15:42
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@Eric I’d recommend Aluffi’s Algebra: Chapter 0. Given some of the other questions you’ve asked on this site, that book might be helpful. – Santana Afton Apr 25 '19 at 15:43
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@SantanaAfton Thanks. I'll definitely check this! – Eric Apr 25 '19 at 15:50