# Isotropy group

The set $G_x$ of elements of a given group $G$, acting on a set $M$ as a group of transformations, that leave a point fixed. This set turns out to be a subgroup of $G$ and is called the isotropy group of the point $x$. The following terminology is used with the same meaning: stationary subgroup, stabilizer, $G$-centralizer. If $M$ is a Hausdorff space and $G$ is a topological group acting continuously on $M$, then $G_x$ is a closed subgroup. If, furthermore, $M$ and $G$ are locally compact, $G$ has a countable base and acts transitively on $M$, then there exists a unique homeomorphism from $M$ into the quotient space $G/H$, where $H$ is one of the isotropy groups; all the $G_x$, $x\in M$, are isomorphic to $H$.
Let $M$ be a smooth manifold and $G$ a Lie group acting smoothly on $M$. Then the isotropy group $G_x$ of a point $x\in M$ induces a group of linear transformations of the tangent vector space $T_x(M)$; the latter is called the linear isotropy group at $x$. On passing to tangent spaces of higher order at the point $x$ one obtains natural representations of the isotropy group in the structure groups of the corresponding tangent bundles of higher order; they are called the higher-order isotropy groups (see also Isotropy representation).