我们和过去永远绝缘了… 而这,在我看来,本身就是一种成就。

If GG is a group with identity element ee, and XX is a set, then a (left) group action α\alpha of GG on XX is a function

α:G×XX,\alpha:G\times X\to X,

that satisfying the following two axioms:

  • Identity: α(e,x)=x\alpha(e,x)=x;
  • Compatibility: α(g,α(h,x))=α(gh,x)\alpha(g,\alpha(h,x))=\alpha(gh,x).
    where α(g,x)\alpha(g,x) is often shortened to gxg\cdot x.

The action of GG on XX is called transitive if for any two points x,yXx,y\in X, there exists a gGg\in G such that gx=yg\cdot x=y.(联通性)

Consider a group GG acting on a set XX. The orbit of an element xXx\in X is the set of element in XX to which xx can be moved by the elements of GG.
The orbit of xx is denoted by GxG\cdot x:

Gx={gx:gG}.G\cdot x=\{g\cdot x:g\in G\}.

实际上 group action 诱导出了一个XX 上的等价关系:xyx\sim y 当且仅当gG, gx=y\exists g\in G,\ g\cdot x=y,也就是说x,yx,y 是联通的。
因此,orbits GxG\cdot x 就构成了XX 的一个划分:要么Gx=GyG\cdot x=G\cdot y,要么GxGy=G\cdot x\cap G\cdot y=\emptyset
而且不难发现,group action 是 transitive 的当且仅当只有一个 orbit:xX, Gx=X\forall x\in X,\ G\cdot x=X

The set of all orbits of XX under the action of GG is written as X/GX/G, and is called the quotient of the action.

In geometric situations it may be called the orbit space, while in algebraic situations it may be called the space of coinvariants, and written XGX_G, by contrast with the invariants (fixed points), denoted XGX^G: the coinvariants are a quotient while the invariants are a subset.

A subset YXY\subseteq X is said to be invariant under GG if GY={gy:gG,yY}=YG\cdot Y=\{g\cdot y:g\in G,y\in Y\}=Y, which is equivalent to GYYG\cdot Y\subseteq Y.
Moreover, YY is said to be fixed under GG if gy=yg\cdot y=y for all gG,yYg\in G,y\in Y.

Every orbit is an invariant subset of XX on which GG acts transitively. Conversely, any invariant subset of XX is a union of orbits.

Given gG,xXg\in G,x\in X with gx=xg\cdot x= x, it is said that xx is a fixed point of gg or gg fixes xx.
For every xXx\in X, the stabilizer subgroup of GG with respect to xx (also called the isotropy group or little group) is the set of all elements in GG that fixes xx:

Gx={gG:gx=x}.G_x=\{g\in G:g\cdot x=x\}.

This is a subgroup of GG, though typically not a normal one.
The action of GG on XX is free if and only if all stabilizers are trivial.

The kernel NN of the homomorphism with the symmetric group, GSym(X)G \to Sym(X), is given by the intersection of the stabilizers GxG_x for all xXx\in X. If NN is trivial, the action is said to be faithful (or effective).

Let x,yXx,y\in X such that y=gxy=g\cdot x for some gGg\in G, then the two stabilizer groups are related by Gy=gGxg1G_y=gG_xg^{-1}.

The above says that the stabilizers of elements in the same orbit are conjugate to each other. Thus, to each orbit, we can associate a conjugacy class of a subgroup of GG (that is, the set of all conjugates of the subgroup). Let (H)(H) denote the conjugacy class of HH. Then the orbit OO has type (H)(H) if the stabilizer GxG_x of some/any xx in OO belongs to (H)(H). A maximal orbit type is often called a principal orbit type.

(Orbit-stabilizer Theorem)
Orbits and stabilizers are closely related. For a fixed xx in XX, consider the map f:GXf : G \to X given by ggxg \mapsto g\cdot x. By definition the image f(G)f(G) of this map is the orbit GxG\cdot x. The condition for two elements to have the same image is

f(g)=f(h)    gx=hx    g1hx=x    g1hGx    hgGx.f(g)=f(h)\iff g\cdot x=h\cdot x\iff g^{-1}h\cdot x=x\iff g^{-1}h\in G_x\iff h\in gG_x.

In other words, f(g)=f(h)f(g) = f(h) if and only if gg and hh lie in the same coset for the stabilizer subgroup GxG_x.
Thus, the fiber f1({y})f^{−1}(\{y\}) of ff over any yy in GxG\cdot x is contained in such a coset, and every such coset also occurs as a fiber.
Therefore ff induces a bijection between the set G/GxG / G_x of cosets for the stabilizer subgroup and the orbit GxG\cdot x, which sends gGxgxgG_x \mapsto g\cdot x. This result is known as the orbit-stabilizer theorem.

If GG is finite then the orbit-stabilizer theorem, together with Lagrange’s theorem, gives

Gx=[G:Gx]=G/Gx.|G\cdot x|=[G:G_x]=|G|/|G_x|.

Example: Let GG be a group of prime order pp acting on a set XX with kk elements.
Since each orbit has either 11 or pp elements(因为素数阶群必然是循环群), there are at least kmodpk\mod p orbits of length 11(因为 orbits 构成XX 的一个划分,而且 orbits 大小要么是pp 要么是11,所以至少有Xmodp|X|\mod p 个 orbits 大小为 1), which are GG-invariant elements. More specifically, kk and the number of GG-invariant elements are congruent modulo pp.
这个结果特别有用,因为它可以用于计数论证(通常在XX 也是有限的情况下)。

Example: We can use the orbit-stabilizer theorem to count the automorphisms of a graph.
Consider the cubical graph as pictured below, and let GG denote its automorphism group. Then GG acts on the set of vertices {1,2,3,...,8}\{1,2,3,...,8\}, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, G=G1G1=8G1|G|=|G\cdot 1||G_1|=8|G_1|(因为 action 是 transitive 的,所以G1={1,2,...,8}G\cdot 1=\{1,2,...,8\}). Applying the theorem now to the stabilizer G1G_1, we can obtain G1=G12(G1)2|G_1| = |G_1 \cdot 2| |(G_1)_2|. Any element of GG that fixes 11 must send 22 to either 22, 44, or 55. Thus, G12=3|G_1\cdot 2|=3(因为G12={2,4,5}G_1\cdot 2=\{2,4,5\}). Applying the theorem a third time gives (G1)2=(G1)23((G1)2)3|(G_1)_2|=|(G_1)_2\cdot 3||((G_1)_2)_3|. Any element of GG that fixes 1,21,2 must send 33 to either 33 or 66(譬如关于平面 1278 对称,自同构群不是旋转群,它不需要保持手性,而只需要保持顶点间的连边就可以了), thus (G1)23=2|(G_1)_2\cdot 3|=2. One also sees that ((G1)2)3((G_1)_2)_3 consists only of the identity automorphism, as any element of GG fixing 1,21, 2 and 33 must also fix all other vertices, since they are determined by their adjacency to 11, 22 and 33. Combining the preceding calculations, we can now obtain G=8321=48|G| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48.

(Burnside’s lemma)

X/G=1GgGXg,|X/G|=\frac{1}{|G|}\sum_{g\in G}|X^g|,

where Xg={xX:gx=x}X^g=\{x\in X:g\cdot x=x\} is the set of elements fixed by gg.

This result is mainly of use when GG and XX are finite, when it can be interpreted as follows: the number of orbits is equal to the average number of points fixed per group element.