Isotropy group

Fixed points and stabilizer subgroups

Given ${\displaystyle g}$ in ${\displaystyle G}$ and ${\displaystyle x}$ in ${\displaystyle X}$ with ${\displaystyle g\cdot x=x}$, it is said that "${\displaystyle x}$ is a fixed point of ${\displaystyle g}$" or that "${\displaystyle g}$ fixes ${\displaystyle x}$". For every ${\displaystyle x}$ in ${\displaystyle X}$, the stabilizer subgroup of ${\displaystyle G}$ with respect to ${\displaystyle x}$ (also called the isotropy group or little group) is the set of all elements in ${\displaystyle G}$ that fix ${\displaystyle x}$: $${\displaystyle G_{x}=\{g\in G:g\cdot x=x\}.}$$ This is a subgroup of ${\displaystyle G}$, though typically not a normal one. The action of ${\displaystyle G}$ on ${\displaystyle X}$ is free if and only if all stabilizers are trivial. The kernel ${\displaystyle N}$ of the homomorphism with the symmetric group, ${\displaystyle G\rightarrow \operatorname {Sym} (X)}$, is given by the intersection of the stabilizers ${\displaystyle G_{x}}$ for all ${\displaystyle x}$ in ${\displaystyle X}$. If ${\displaystyle N}$ is trivial, the action is said to be faithful (or effective).