# Visualization of groups with a normal subgroup_rev#1

Let $$G$$ be a group and $$H \unlhd G$$. In general, $$H=H_Z \sqcup H_{G \setminus Z}$$, where $$H_Z:=H \cap Z(G)$$ and $$H_{G \setminus Z}:=H \cap (G \setminus Z(G))$$. I'm investigating on a plausible visual model for the pair $$(G,H)$$. I'll provisionally retain the following one, till some inconsistency will pop up for revision/reject. By "inconsistency" I mean either a contradiction with, or the inability to show, known algebraic facts.

1. $$G$$ is the euclidean 3-space and $$e$$ its (geometrical) center;
2. given $$g \in G \setminus Z(G)$$, the centralizer $$C_G(g)$$ is a ball whose poles are $$g$$ and the element $$g_{\operatorname{op}}$$, opposite to $$g$$ with respect to $$e$$ and distant $$\mathtt{r}_Z$$ from $$e$$;
3. by 2, $$Z(G)=\bigcap_{g \in G}C_G(g)$$ is the ball centered in $$e$$ of radius $$\mathtt{r}_Z$$;
4. given $$g \in G$$, the right cosets $$C_G(g)g'$$, $$g' \in G$$, are eccentric, thick "shells" embedding $$C_G(g)$$ ("onion"-like); the eccentricity gives the possibility to single out as another partition of $$G$$ the one made of the left cosets; shell's average thickness is a decresing function of shell's size, so as to get a hint of the bijection between any pair of cosets (constant volume);
5. $$\forall h \in H_Z$$, the conjugacy orbit by $$h$$ is pointwise, being $$O_h=\lbrace g^{-1}hg, g \in G \rbrace = \lbrace h \rbrace$$;
6. by 5, $$H_Z$$ is an axis of $$Z(G)$$ (or anything topologically equivalent to that);
7. once popped out of $$Z(G)$$, conjugacy orbits become real ones, namely circles around the axis induced by $$H_Z$$, which globally form a "polar" toroidal surface, embedding $$Z(G)$$ (this is $$H_{G \setminus Z}$$);
8. $$H$$ splits $$G \setminus H$$ into two regions: an "inner" one and an "outer" one, say $$G \setminus H = G_{H}$$; given $$g \in G_{, the coset $$Hg$$ is the toroidal surface by $$g$$, slicing $$Z(G)$$; given $$g' \in G_{>H}$$, the coset $$Hg'$$ is the surface by $$g'$$, embedding $$H$$ and topologically equivalent to a 2-sphere.

This model is possibly still far from the "reality", but maybe we can get a better one by addressing some points raised by this one:

#1. Does $$H$$'s closure have some algebraic validity? What would it mean?

#2. Would the special case $$Z(G)=\lbrace e \rbrace$$ be consistently described by the above model, i.e. with $$Z(G)$$ "deflated" down to one point?

#3. Given $$h \in H_{G \setminus Z}$$, are the algebraic loci $$C_G(h) \cap H$$ and $$C_G(h) \cap O_h$$ suitably accounted for in terms of the sphere/torus crossing expected from the model?

• Great question! (+1) – Shaun Mar 19 at 16:27
• I mean: I don't follow it exactly as it might be beyond me but it looks very interesting, certainly; what I understand of it looks alright to me. – Shaun Mar 19 at 16:32
• How did you make the diagrams (i.e., what software did you use)? – Shaun Mar 19 at 16:41
• Just MS Office shapes package (Word, PowerPoint,...). – Luca Mar 19 at 16:46
• I can't make head or tail of this. What is $g_{op}$? What is $r_Z$? etc. Could you explain what your model looks like for, say, $S_3$? (The smallest non-abelian group.) – verret Mar 20 at 6:58

I am reading the question as you are asking if there is a group $$G$$ and a subgroup $$H$$ that satisfy the above properties. I haven't read through all of your properties, but it is already impossible with properties 2, 3.
Namely, say you have a group $$G$$ which can be identified with $$\mathbb R^3$$. Your property 3 says $$Z(G)$$ is a ball of some radius, say $$r_Z$$, about $$e$$. (You say sphere, but I assume you mean ball, because certainly you need $$e \in Z(G)$$. I will assume the same for property 2.)
Pick any $$g \in G - Z(G)$$. Then property 2 says $$C_G(g)$$ is a ball of some radius $$r > r_e$$ about $$e$$. Take some $$h$$ in $$C_G(g) - Z(G)$$ in the interior of this ball. Then $$C_G(h)$$ is a ball of radius $$< r$$ about $$e$$, but $$gh=hg$$ means $$g \in C_g(h)$$. However $$g$$ has distance $$r$$ from $$e$$, a contradiction.