# Intuition of Picture - Collapse, Factor Group, Homomorphism, Normal Subgroup - Fraleigh p. 144 Figure 15.1

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.

(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?

• To understand this to the fullest, you should know what 'equivalent relation', and 'Equivalence class' are. Do you know them? – user49685 Feb 19 '14 at 18:31
• It's kind of a crappy picture. – Alexander Gruber Feb 22 '14 at 7:52
• @AlexanderGruber: Can you please flesh out why? – Group Theory Feb 26 '14 at 7:35
• @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, – Alexander Gruber Feb 26 '14 at 8:28
• representing the action of $\gamma$) onto dots in $G/N$ (the coset elements of the factor group). – Alexander Gruber Feb 26 '14 at 8:30