# orbit-stabilizer theorem

Suppose that $G$ is a group acting (http://planetmath.org/GroupAction) on a set $X$. For each $x\in X$, let $Gx$ be the orbit of $x$, let $G_{x}$ be the stabilizer of $x$, and let ${\cal L}_{x}$ be the set of left cosets of $G_{x}$. Then for each $x\in X$ the function $f\colon Gx\to{\cal L}_{x}$ defined by $gx\mapsto gG_{x}$ is a bijection. In particular,

 $|Gx|=[G:G_{x}]$

and

 $|Gx|\cdot|G_{x}|=|G|$

for all $x\in X$.

Proof:
If $y\in Gx$ is such that $y=g_{1}x=g_{2}x$ for some $g_{1},g_{2}\in G$, then we have $g_{2}^{-1}g_{1}x=g_{2}^{-1}g_{2}x=1x=x$, and so $g_{2}^{-1}g_{1}\in G_{x}$, and therefore $g_{1}G_{x}=g_{2}G_{x}$. This shows that $f$ is well-defined.

It is clear that $f$ is surjective. If $gG_{x}=g^{\prime}G_{x}$, then $g=g^{\prime}h$ for some $h\in G_{x}$, and so $gx=(g^{\prime}h)x=g^{\prime}(hx)=g^{\prime}x$. Thus $f$ is also injective.

Title orbit-stabilizer theorem OrbitstabilizerTheorem 2013-03-22 12:23:10 2013-03-22 12:23:10 yark (2760) yark (2760) 22 yark (2760) Theorem msc 20M30