# Visualization of groups with a normal subgroup

Suppose $$G$$ a group and $$H \triangleleft G$$ (proper normal subgroup). The simplest way to visualize this basic setup is that (Venn-wise) of a bubble ($$H$$) into a bigger one ($$G$$), sharing the unit and more.

Now, any given $$h \in H$$ determines an equivalence class of $$H$$ by conjugacy, namely $$O_h:=\lbrace g'^{-1}hg', g' \in G\rbrace$$, and -for any $$g \in G$$- an equivalence class of $$G$$, namely $$C_G(h)g$$ (right coset by $$g$$ of the centralizer of $$h$$ in $$G$$).

Is there any "topologically coherent" visualization of the two "manifolds" $$O_h$$ and $$C_G(h)g$$ in the naïve picture of $$G$$ and $$H$$ above?

## Addendum

I've realized that the naïve setup in the opening is actually inconsistent: $$G$$, its "slicing" (or "foliation") into "sheets" (or "shells") $$Hg$$, and $$H$$'s partitioning into "fibers" or "orbits" $$O_h$$, must have "least dimension" 3, 2 and 1, respectively. Therefore, my question is rather turned into an investigation on the openness/closure of $$Hg$$'s and $$C_G(h)g$$'s in $$G$$ and $$O_h$$'s in $$H$$, and on their mutual intersections (if any).

## 1 Answer

In my question, "topologically coherent" meant not contradicting -and possibly visually hinting- algebraic facts. It seems to me that the model 1$$\div$$12 hereafter is coherent for a group $$G$$ which fixes the conjugacy classes of $$H$$ under conjugation (such that $$C_G(h)g \cap H \ne \emptyset, \forall h \in H, g \in G$$), or perhaps even a specialization of it (see item 10):

1. $$G$$ is a sphere, and the unit $$e$$ its center;
2. $$H$$ is the equatorial cut of $$G$$;
3. $$Hg$$ is the $$\phi$$-latitude cut of $$G$$, $$\phi$$ being the latitude of $$g \in G$$;
4. the bijection between each pair of cosets of $$H$$ is represented by the bundle of pole-to-pole fibers throughout $$G$$ (not shown in the picture);
5. $$O_h$$ is the circle by $$h$$ in the equatorial plane;
6. call meridian segment the region of $$G$$ induced by a meridian half-plane (i.e. a half-plane through the axis); the Orbit-Stabilizer bijection $$\chi$$ (see here) is represented by the single crossing of each meridian segment with $$O_h$$;
7. $$C_G(h)g$$ is the meridian segment containing $$g$$ and open on the axis (since $$e \notin C_G(h)g$$);
8. $$C_G(h)$$ is the meridian segment containing $$h$$; it is the only one closed on the axis (since $$e \in C_G(h)$$);
9. $$C_G(h) \cap H$$ is the equatorial radial fiber by $$h$$ ($$\alpha$$ in the picture);
10. $$C_G(h)g \cap H$$ is the equatorial radial fiber by $$\chi(C_G(h)g)$$ ($$\beta$$ in the picture);
11. $$C_G(h)g \cap Hg$$ is the "axially radial"(?) fiber by $$g$$ ($$\gamma$$ in the picture);
12. $$Z(G)$$ is contained in the axis, since this latter represents the subgroup $$C_G:=\bigcap_{h \in H}C_G(h)$$ (see item 8) and $$Z(G)=\bigcap_{g \in G}C_G(g)=C_G \cap \left( \bigcap_{g \in \complement_G(H)}C_G(g) \right) \subseteq C_G$$

The following picture shows the whole. Do you see any contradiction or inconsistency?

Do you see other algebraic facts that I could add to the model as visual feature?

Do you see some "morphing" of the above model into one representing more general groups?