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 $\mathtt{r}_Z$; enter image description here
  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); enter image description here
  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 $G \setminus H = G_{<H} \sqcup G_{>H}$; given $g \in G_{<H}$, 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. enter image description here

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?

  • $\begingroup$ Great question! (+1) $\endgroup$ – Shaun Mar 19 at 16:27
  • $\begingroup$ 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. $\endgroup$ – Shaun Mar 19 at 16:32
  • $\begingroup$ How did you make the diagrams (i.e., what software did you use)? $\endgroup$ – Shaun Mar 19 at 16:41
  • $\begingroup$ Just MS Office shapes package (Word, PowerPoint,...). $\endgroup$ – Luca Mar 19 at 16:46
  • $\begingroup$ 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.) $\endgroup$ – 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.


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