OrbitStabilizerTheorem 2002-02-19 06:41:46 2008-05-09 06:37:57 Theorem \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} Suppose that $G$ is a group \PMlinkname{acting}{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 $x\in X$. {\bf Proof}:\\ If $y\in Gx$ is such that $y=g_1x=g_2x$ for some $g_1,g_2\in G$, then we have $g_2^{-1}g_1x=g_2^{-1}g_2x=1x=x$, and so $g_2^{-1}g_1\in G_x$, and therefore $g_1G_x=g_2G_x$. This shows that $f$ is well-defined. It is clear that $f$ is surjective. If $gG_x = g'G_x$, then $g = g'h$ for some $h \in G_x$, and so $gx = (g'h)x= g'(hx) = g'x$. Thus $f$ is also injective.