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Let $N \unlhd G$. In the factor group $G/N$, the subgroup $N$ acts as identity element. Regard N as being collapsed to a single element, to the identity element. This collapsing of N together with the algebraic structure of G require that other subsets of G, namely the cosets of N, also collapse into a single element in the factor group. This collapsing can be seen in Fig. 15.1 underneath.

Recall from Theorem 14.9: $\phi: G \to G/N$ defined by $\phi(g) =gN$ for $g \in G$ is a onto homomorphism. Figure 15.1 resembles Fig. 13.14. But in Fig. 15.1, the image group under the homomorphism is actually formed from G. We can view the "line" $G/N$ at the bottom of the figure as obtained by collapsing to a point each coset of N in another copy of G. Each point of $G/N$ thus corresponds to a whole vertical line segment in the shaded portion, representing a coset of N in G. Remember that multiplication of cosets in $G/N$ can be computed by multiplying in G, using any representative elements of the cosets as shown.

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(1.) Could someone please flesh out the intuition of the text? What's this picture trying to unfold?
What are the nubs? I know what is group G, factor group G/N, identity e, homomorphism $\phi$.

(2.) I'm perplexed by all the collapses and collapsing. Why are you authorized to do this?
Why are you authorized to change your group like this?

(3.) How is a subgroup N collapsed to a single element, to the identity element'? Why want this?
Factor groups usually require $N \neq \{id\}$, because $G/\{id\} \cong G$.
But this doesn't unfold anything about $G$? Did I muff something?

(4.) What do the dotted lines mean?

(5.) I can see this looks like Fig 13.14. But can someone please flesh out the similarities and differences? What are the nubs?

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  • $\begingroup$ To understand this to the fullest, you should know what 'equivalent relation', and 'Equivalence class' are. Do you know them? $\endgroup$ – user49685 Feb 19 '14 at 18:31
  • $\begingroup$ It's kind of a crappy picture. $\endgroup$ – Alexander Gruber Feb 22 '14 at 7:52
  • $\begingroup$ @AlexanderGruber: Can you please flesh out why? $\endgroup$ – Group Theory Feb 26 '14 at 7:35
  • $\begingroup$ @FrankMuer Many of your questions are answered by any standard treatment of quotient groups. I don't think I could exposit better than most books can. What I can say is that this picture is illustrates the relationship between multiplication in $G$ with the multiplication of cosets in $G/N$, via the projection homomorphism $\gamma$. The idea is to show that $\gamma$ sends whole sets of elements in $G$ to the one element of $G/N$, in such a way that $\gamma$ respects with the group operation. In this way these sets (represented by the green lines) are "projected" (via dotted lines, $\endgroup$ – Alexander Gruber Feb 26 '14 at 8:28
  • $\begingroup$ representing the action of $\gamma$) onto dots in $G/N$ (the coset elements of the factor group). $\endgroup$ – Alexander Gruber Feb 26 '14 at 8:30

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