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Two group actions on $S_1,S_2$ are equivalent if there is a bijection $\varphi :S_1\to S_2$ such that $\varphi (gs)=g\varphi (s)$ for all $g,s$. Such a bijection is equivariant.

The orbit-stabilizer theorem in this form says that the action on $O_s$ is equivalent to the left coset action on $G/G_s$, by $g{G}_s\mapsto gs$. This map is well-defined and injective because $gs=hs$ exactly when $h^{-1}g\in G_s$, and it is onto by the definition of the orbit. It is equivariant by multiplication.

Thus every transitive action is a coset action. Two such actions $G/H$ and $G/K$ are equivalent exactly when $H,K$ are conjugate: stabilizers at different points of one orbit are conjugate, and a conjugacy explicitly supplies a coset-action equivalence. This last classification is what makes counting by orbit types possible.

Two group actions on $S_1,S_2$ are equivalent if there is a bijection $\varphi :S_1\to S_2$ such that $\varphi (gs)=g\varphi (s)$ for all $g,s$. Such a bijection is equivariant.

The orbit-stabilizer theorem in this form says that the action on $O_s$ is equivalent to the left coset action on $G/G_s$, by $g{G}_s\mapsto gs$. This map is well-defined and injective because $gs=hs$ exactly when $h^{-1}g\in G_s$, and it is onto by the definition of the orbit. It is equivariant by multiplication.

Thus every transitive action is a coset action. Two such actions $G/H$ and $G/K$ are equivalent exactly when $H,K$ are conjugate: stabilizers at different points of one orbit are conjugate, and a conjugacy explicitly supplies a coset-action equivalence. This last classification is what makes counting by orbit types possible.

The hypothesis is a two-transitive group action. Given $\alpha ,\beta \ne {\omega }_1$, apply it to $({\omega }_1,\alpha )$ and $({\omega }_1,\beta )$. The resulting element fixes ${\omega }_1$ and sends $\alpha$ to $\beta$, so the stabilizer subgroup acts transitively on the remaining $n-1$ points.

The action on all of $\Omega$ is also transitive. By the orbit-stabilizer theorem,Thus $\boxed{n(n-1)\mid |G|}$. The diagonal pairs form one orbit, while two-transitivity makes all ordered unequal pairs a second orbit. Equality is preserved by a group action, so these cannot mix. There are exactly two orbits on $\Omega \times \Omega$.

For a group action, the orbit and stabilizer areThe orbit-stabilizer theorem for a finite group states

For a regular tetrahedron, rotations fixing a vertex rotate about the line from that vertex through the opposite face centre. Their angles are $0,2\pi /3,4\pi /3$, so the stabilizer is a cyclic group of order three. Rotations act transitively on the four vertices, giving $|G_1|=4\cdot 3=12$ by the orbit-stabilizer theorem.

For a cube, the corresponding axis is the body diagonal through the fixed vertex and its opposite. The three incident edges can be cyclically permuted by the same three angles, so again the stabilizer is cyclic of order three. The orbit has eight vertices. Hence

Any nonzero vector in ${\mathbb{F}}_2^3$ extends to a basis. Sending one such basis to another gives an element of the general linear group, so its group action on the seven nonzero vectors is a transitive group action. Counting successive independent columns givesBy the orbit-stabilizer theorem, the stabilizer $H$ of a chosen nonzero vector $v$ has order $168/7=24$.

There are three two-dimensional linear subspaces containing $v$, corresponding to the three one-dimensional linear subspaces of ${\mathbb{F}}_2^3/⟨v⟩$. The stabilizer $H$ acts transitively on them: extend a basis of either plane beginning with $v$ to a basis of the whole vector space. Thus the stabilizer of a flag $⟨v⟩<P$ has order $24/3=8$.

If $Q\le G$ has order eight, each orbit of a group action of $Q$ on the seven nonzero vectors has size a power of two. Since seven is odd, at least one orbit of a group action has size one. Thus $Q$ fixes a nonzero vector $v$. Its group action on the three planes containing $v$ likewise has a fixed plane $P$. Therefore $Q$ is contained in, and by order equals, the stabilizer of that flag. The group action of $G$ is transitive on these flags, so all such subgroups are conjugate subgroups.

Distinct flags give distinct stabilizers. In a basis adapted to a flag, its order-eight stabilizer consists ofIts only common fixed nonzero vector is $e_1$. Its only invariant plane is $⟨e_1,e_2⟩$: an invariant plane is the kernel of a linear map of a nonzero linear functional, and over ${\mathbb{F}}_2$ that linear functional must be fixed; the displayed matrices have only the fixed nonzero linear functional $e_3^{\ast }$. Hence each subgroup specifies its flag uniquely. There are seven choices of $v$ and three choices of $P$, giving

Relative to the three successive blocks of sizes $1,2,2$, the stabilizer consists of the block upper triangular matricesThe off-diagonal blocks contain $2+2+4=8$ arbitrary field entries. Thus its order is $(q-1)|\mathrm{GL}(2,q){|}^2{q}^8$. By orbit-stabilizer theorem,Equivalently, choose the line first and then a two-dimensional subspace of the four-dimensional quotient:This also checks the stabilizer count independently.

A regular icosahedron is a Platonic solid with twenty equilateral triangular faces, twelve vertices and thirty edges. Five faces meet at each vertex. Its orientation-preserving rotational symmetry has order sixty: the orbit-stabilizer theorem gives twenty faces times the three rotations stabilizing a chosen face.

A group action on a nonempty set $X$ is simply transitive when exactly one group element sends any chosen point to any other:Equivalently it is both a transitive group action and a free group action; this is also a regular group action. Transitivity supplies a transport, and freeness makes it unique. Conversely, putting $y=x$ in the uniqueness condition shows that every stabilizer subgroup is $\{e\}$, since the identity element already fixes $x$. In a transitive group action, checking that just one stabilizer subgroup is trivial suffices: all the others are conjugate subgroups.

The cyclic group $C_3$ in the diagram under transitive group action is simply transitive. More generally, the left regular action of any group $G$ on itself is simply transitive: the equation $gx=y$ has the unique solution $g=y{x}^{-1}$, using the inverse element of $x$. This shows why the property is useful: every position has a unique label relative to a chosen starting position.

Choose a point $x_0\in X$. Its orbit mapis a bijection and an equivariant map from the left regular action: ${\varphi }_{x_0}(hg)=h\cdot {\varphi }_{x_0}(g)$. This identifies $X$ with a copy of $G$, but $X$ does not come with a preferred identity element. If $x_1=a\cdot x_0$, then ${\varphi }_{x_1}(g)={\varphi }_{x_0}(ga)$: changing the chosen point composes this identification with right multiplication by $a$. An old label $g$ therefore becomes the new label $g{a}^{-1}$, which also specifies the side of multiplication for a noncommutative group.

The transport from $x$ to $y$ remains independent of the chosen point. Writing $x=b\cdot x_0$ and $y=c\cdot x_0$, it is $c{b}^{-1}$. With $x_1=a\cdot x_0$, the new labels are $b{a}^{-1}$ and $c{a}^{-1}$, andFor a finite group, simple transitivity implies $|X|=|G|$. Conversely, a transitive group action with $|X|=|G|<\infty$ is simply transitive by the orbit-stabilizer theorem. Equality of cardinalities alone is insufficient without transitivity: the cyclic group $C_2$ can fix both points of a two-point set, giving two group orbits and two possible self-transports at each point.

For a centered static Skyrmion $U_0$, the effective rotation-isorotation group is $G=SO(3{)}_{\mathrm{space}}\times SO(3{)}_{\mathrm{iso}}$. Its stabilizer subgroup $H$ consists of pairs obeying $A{U}_0(R^{-1}x)A^{†}=U_0(x)$. Thus distinct rigid orientations form the homogeneous space $G/H$, by the orbit-stabilizer theorem. The unit hedgehog has diagonal $SO(3)$ stabilizer and three orientation coordinates; the toroidal two-Skyrmion has a one-dimensional continuous stabilizer and five; the tetrahedral and cubic cases have discrete stabilizers and six. Adding three translations gives zero-mode orbit dimensions 6, 8, 9, 9. These are symmetry-generated low-energy coordinates, not a claim that non-Bogomolny multi-Skyrmion solutions possess an exact flat moduli space of arbitrary separations.

A matrix $\left(\begin{array}{cc}a& b\\ c& d\end{array}\right)$ acts on the projective line by the Möbius transformation $x\mapsto (ax+b)/(cx+d)$, with the pole sent to infinity and infinity sent to $a/c$ when $c\ne 0$. The action is transitive because $\left(\begin{array}{cc}t& -1\\ 1& 0\end{array}\right)$ sends infinity to any finite $t$. The stabilizer subgroup of infinity consists of $\left(\begin{array}{cc}a& b\\ 0& a^{-1}\end{array}\right)$ with $a\ne 0$. The orbit-stabilizer theorem gives $|{\mathrm{SL}}_2({\mathbb{F}}_q)|=q(q^2-1)$. Its kernel is the scalar matrices of determinant one: a transformation fixing infinity has $c=0$, fixing zero then forces $b=0$, and fixing one gives $a=d$. Thus the kernel is $\{I,-I\}$, with these matrices coinciding in characteristic two.

Let a separable polynomial $f$ of degree four be irreducible over a field $F$. If all three pair sums $r_1={\alpha }_1{\alpha }_2+{\alpha }_3{\alpha }_4$, $r_2={\alpha }_1{\alpha }_3+{\alpha }_2{\alpha }_4$, and $r_3={\alpha }_1{\alpha }_4+{\alpha }_2{\alpha }_3$ lie in $F$, its Galois group is the Klein four-group. Indeed $r_1-r_2=({\alpha }_1-{\alpha }_4)({\alpha }_2-{\alpha }_3)\ne 0$, and similarly the other pair sums differ. Every Galois automorphism therefore fixes each of the three partitions into pairs. The kernel of this action of $S_4$ on three partitions is $V_4=\{1,(12)(34),(13)(24),(14)(23)\}$. Irreducibility gives a transitive group action on the four roots, so the orbit-stabilizer theorem forces $4$ to divide the group order. Hence the Galois group equals $V_4$. The argument also works in characteristic two.

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 cube vertices $(\pm 1,\pm 1,\pm 1)$, the four vertices of coordinate product $+1$ and the four of product $-1$ form the two inscribed regular tetrahedra. A cube rotation is a signed permutation matrix of determinant one and maps each product class to one class. A quarter-turn about a coordinate axis exchanges the classes. The orbit of either tetrahedron therefore has size two, while the cube rotation group has order 24. The orbit-stabilizer theorem gives its setwise stabilizer subgroup order 12; this is the tetrahedron's proper rotation group.

A group action on a nonempty set $X$ is transitive when every two points lie in the same orbit of a group action, equivalently when for all $x,y\in X$ there is a group element $g$ with $g\cdot x=y$. There is exactly one group orbit: every position is reachable from every other. Requiring $X\ne \varnothing$ avoids calling the group action on the empty set transitive merely because the quantified statement has no points to test.^[ref]

Here a left group action of a group $G$ means a function $G\times X\to X$, written $(g,x)\mapsto g\cdot x$, satisfyingwhere $e$ is the identity element. Products move the point by $h$ first, then by $g$. Transitivity says that an orbit map $G\to X$, $g\mapsto g\cdot x$, is surjective for every $x\in X$; it suffices to check one chosen point.^[ref][ref]

For example, the symmetric group $S_3$ acts on $X=\{1,2,3\}$ by $\sigma \cdot i=\sigma (i)$. This is a transitive group action, but it is not a free group action: the stabilizer subgroup of $1$ is $\{e,(23)\}$. Both the transposition $(12)$ and the cyclic permutation $(123)$ send $1$ to $2$. In contrast, the cyclic group $C_3=\mathbb{Z}/3\mathbb{Z}$ acts on itself by $k\cdot x=k+x(\mathrm{mod}3)$, using modular arithmetic. The group element sending $x$ to $y$ is uniquely $k=y-x(\mathrm{mod}3)$. These two group actions reach the same number of points, but only the second has unique transport between them.

The stabilizer subgroup $G_x=\{h\in G:h\cdot x=x\}$ measures the ambiguity of transport. If $g_0\cdot x=y$, thenThe first set is a left coset, and the second identity says that stabilizer subgroups in one group orbit are conjugate subgroups. Briefly, $g\cdot x=g_0\cdot x$ exactly when $g_0^{-1}g\in G_x$. Thus, for a finite group in a transitive group action, the orbit-stabilizer theorem gives $|G|=|X||G_x|$, and there are exactly $|G_x|$ transports for each pair of points. The example counts are $6=3\cdot 2$ for $S_3$ and $3=3\cdot 1$ for $C_3$. This separates the number of reachable positions from the number of ways to reach each one.^[ref]