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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 Answers1

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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.

enter image description here

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?