If $a, b$ at the same orbit, then $\operatorname{stab}(a) \cong \operatorname{stab}(b)$ $\newcommand{\stab}{\operatorname{stab}}$Group $G$ acts on set $X$. Let $a, b \in X$ be at the same orbit. Prove $\stab(a) \cong \stab(b)$ Any hints?

Just write it out.

Step 1

> $a$ and $b$ are in the same orbit iff there is $g \in G$ such that $b = a^{g}$.

Step 2

> $x$ fixes $a$ iff $a^{x} = a$.

Step 3

> $y$ fixes $b$ iff $a^{g} = b = b^{y} = a^{gy}$ iff $g y g^{-1}\ldots$