Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics I / 7 / a

A left group action of $G$ on a set $S$ is a function $G\times S\to S$, written $(g,s)\mapsto g\cdot s$, satisfyingfor all $g,h\in G$, $s\in S$. The orbit of a group action through $s$ and the stabilizer subgroup of $s$ are, respectively,The identity element belongs to $G_s$. If $g,h\in G_s$, then $(gh)\cdot s=g\cdot (h\cdot s)=g\cdot s=s$, so $gh\in G_s$. If $g\in G_s$, applying $g^{-1}$ to $g\cdot s=s$ gives $s=g^{-1}\cdot s$, so $g^{-1}\in G_s$. These facts prove that the stabilizer subgroup is a subgroup of $G$.

The orbit-stabilizer theorem identifies the orbit of a group action of $s$ with the set of left cosets of $G_s$. Explicitly, there is a bijectionHere $G/G_s$ denotes a set of cosets; it is not asserted to be a quotient group. To prove the identification, observe thatThis equivalence proves both well-definedness and injectivity. Every element of the orbit of a group action has the form $g\cdot s$, proving surjectivity.

In particular, for a finite group, the cardinalities satisfyThe last equality follows from Lagrange's theorem applied to the stabilizer subgroup.