A few important aspects of the relationship $H \lhd N_G(H) \le G$ are highlighted in Figure 7.31. First, the size of $N_G(H)$ is some multiple of |H|, and the size of G is some multiple of $N_G(H)$,
Question 1. I know because of Lagrange's Theorem, $H \lhd N_G(H) \le G \implies |H| / |N_G(H)| / |G|$.
But how can you see these 3 relationships from Figure 7.31?
so that the three are only different when H is fairly small compared to G (at most one fourth its size).
Question 2. How can you see this from Figure 7.31? or do I need to refer to some separate proof?
Second, the boundary of $N_G(H)$ falls on the boundaries of left cosets of H, cutting through none of them; it includes cosets whole or not at all, as we saw earlier. Third, it contains exactly those left cosets of H that are also right cosets. Although all of G is full of copies of H (its left cosets), the $N_G(H)$ boundary is as large as it can be while still keeping all of the relationships $H \lhd N_G(H) \le G$ true.
Question 3. Can we say anything more about the white space outside of $N_G(H)$ and its cosets $a_iN_g(H)$? I know it's $G$.
Question 4. I know $a_iN_g(H)$ are cosets of $N_G(H)$? Anything special about them? How do you think about them intuitively?
Question 5. Can someone please elaborate on other important observations from this picture? I feel I'm missing things. I drew the green. Is it ok?
This is from Nathan Carter page 142 143 Visual Group Theory.