Note - Orbits and Stabilizers

notegroup-theory

Orbit (Group) Stabilizer (Group)

Remark

If $(G,\cdot )\circlearrowleft X$, we have an equivalence relation on $X$ $x\sim x^{\prime }$ if $x^{\prime }=g\bullet x$ for some $g\in G$. This is indeed an equivalence relation, and $[x]={\mathrm{orb}}_G(x)$.

Corollary

If $(G,\cdot )\circlearrowleft X$,

1. $X$ is partitioned by distinct orbits.
2. $\mathrm{orb}(y)=\mathrm{orb}(x)⟺y\in \mathrm{orb}(x)$.
3. $x\in \mathrm{orb}(x)$.
4. Either $\mathrm{orb}(y)=\mathrm{orb}(x)$ or $\mathrm{orb}(y)\cap \mathrm{orb}(x)=\varnothing$ in $X$.

Recall

If $G\circlearrowleft G$ by conjugation, $g\in G$, we have $|[g]|=\frac{|G|}{|C_G(g)|}$ (where $C_G(g)$ is the centralizer of $g$ in $G$).

Recall

$x\in C_G(g)$ if $gx=xg⟺xg{x}^{-1}=g⟺x\bullet g=g$ (specifically where $\bullet$ is the action by conjugation).

For arbitrary actions, we have “stabilizers”.

Proposition

For $G\circlearrowleft X$, and $x\in X$. The stabilizer, ${\mathrm{stab}}_G(x)\le G$, is a subgroup.

Proof

$e_g\bullet x=x⟹e_G\in {\mathrm{stab}}_G(x)$ $g,g^{\prime }\in {\mathrm{stab}}_G(x)⟹(g{g}^{\prime })\bullet x=g\bullet (g^{\prime }\bullet x)⟹g{g}^{\prime }\in {\mathrm{stab}}_G(x)$

Say $g\bullet x=x⟹g^{-1}\bullet (g\bullet x)=g^{-1}\bullet x$ and $g^{-1}\bullet (g\bullet x)=(g^{-1}g)\bullet x=e_G\bullet x=x⟹g^{-1}\in {\mathrm{stab}}_G(x).\square$

Remark