2
$\begingroup$

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).

$\endgroup$
2
$\begingroup$

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?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.