2
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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.

$\endgroup$

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.