Stabilizer subgroup ($G_x={\mathrm{Stab}}_G(x)$)

For a group action of a group $G$ on a set $X$, the stabilizer subgroup of $x\in X$ consists of the transformations that leave $x$ fixed:It is a subgroup: the identity element fixes $x$, products of transformations fixing $x$ fix $x$, and the inverse element of a transformation fixing $x$ also fixes $x$. A stabilizer subgroup is a subset of $G$, whereas the orbit of a group action is a subset of $X$. See Ackerman's algebra notes, definition 0.4.1.2.

For the square group actions defined under group action, direct calculation givesThe reflection $s$ fixes $A$ and $C$ individually. The reflection $rs$ preserves the unoriented side of a polygon $AB$ by exchanging its two vertices of a polygon. The half-turn rotation $r^2$ exchanges the two ends of the unoriented diagonal of a polygon $AC$ while preserving that diagonal of a polygon. Thus fixing a side of a polygon or diagonal of a polygon as an object does not mean fixing every point on it; the latter condition uses a pointwise stabilizer. The four stabilizer subgroups have group orders $2,2,4,8$, matching group orbit sizes $4,4,2,1$ through the orbit-stabilizer theorem.

A stabilizer subgroup records the transformations still available after imposing a constraint. Fixing a vertex of a polygon of the square leaves two choices; fixing its centroid leaves all eight. In a faithful group action, a stabilizer subgroup can still contain a nonidentity transformation, as $s\in (D_4{)}_A$ shows: a faithful group action requires that only the identity element fix every point of $X$ simultaneously, not that every individual stabilizer subgroup be trivial. The two-point cyclic group example under group action has both stabilizer subgroups equal to the group kernel $\{0,2\}$ of its associated group homomorphism.

For a group action, the pointwise stabilizer of a set $S$ is the subgroup fixing each of its points individually. It is the intersection of their stabilizer subgroups. This is stronger than merely mapping the set to itself.

For a steady-state equivariant dynamical system, an axial subgroup has a one-dimensional real fixed-point subspace of a group action. A complex axial subgroup for a Hopf problem has a one-dimensional complex fixed-point subspace under the spatial group combined with the common phase action. Distinct conjugacy classes may yield distinct types of bifurcating branches.

For the spatial group combined with the temporal phase circle in an equivariant Hopf problem, a complex axial isotropy subgroup is an actual stabilizer whose fixed-point space is one complex line. Its real dimension is two. One must include the kernel of the combined action when writing the full stabilizer, not just a generator having the right fixed space.

// Target: dynamical-systems.bigb