Group Action, Orbits and Stabilizer

我们和过去永远绝缘了… 而这，在我看来，本身就是一种成就。

The action of $G$ on $X$ is called transitive if for any two points $x,y\in X$, there exists a $g\in G$ such that $g\cdot x=y$.（联通性）

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

The set of all orbits of $X$ under the action of $G$ is written as $X/G$, 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 $X_G$, by contrast with the invariants (fixed points), denoted $X^G$: the coinvariants are a quotient while the invariants are a subset.

A subset $Y\subseteq X$ is said to be invariant under $G$ if $G\cdot Y=\{g\cdot y:g\in G,y\in Y\}=Y$, which is equivalent to $G\cdot Y\subseteq Y$.
Moreover, $Y$ is said to be fixed under $G$ if $g\cdot y=y$ for all $g\in G,y\in Y$.

Every orbit is an invariant subset of $X$ on which $G$ acts transitively. Conversely, any invariant subset of $X$ is a union of orbits.

Given $g\in G,x\in X$ with $g\cdot x= x$, it is said that $x$ is a fixed point of $g$ or $g$ fixes $x$.
For every $x\in X$, the stabilizer subgroup of $G$ with respect to $x$ (also called the isotropy group or little group) is the set of all elements in $G$ that fixes $x$:

$$G_x=\{g\in G:g\cdot x=x\}.$$

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

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

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

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 $G$ (that is, the set of all conjugates of the subgroup). Let $(H)$ denote the conjugacy class of $H$. Then the orbit $O$ has type $(H)$ if the stabilizer $G_x$ of some/any $x$ in $O$ belongs to $(H)$. A maximal orbit type is often called a principal orbit type.

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

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

Example: Let $G$ be a group of prime order $p$ acting on a set $X$ with $k$ elements.
Since each orbit has either $1$ or $p$ elements（因为素数阶群必然是循环群）, there are at least $k\mod p$ orbits of length $1$（因为 orbits 构成$X$ 的一个划分，而且 orbits 大小要么是$p$ 要么是$1$，所以至少有$|X|\mod p$ 个 orbits 大小为 1）, which are $G$-invariant elements. More specifically, $k$ and the number of $G$-invariant elements are congruent modulo $p$.
这个结果特别有用，因为它可以用于计数论证（通常在$X$ 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 $G$ denote its automorphism group. Then $G$ acts on the set of vertices $\{1,2,3,...,8\}$, and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, $|G|=|G\cdot 1||G_1|=8|G_1|$（因为 action 是 transitive 的，所以$G\cdot 1=\{1,2,...,8\}$）. Applying the theorem now to the stabilizer $G_1$, we can obtain $|G_1| = |G_1 \cdot 2| |(G_1)_2|$. Any element of $G$ that fixes $1$ must send $2$ to either $2$, $4$, or $5$. Thus, $|G_1\cdot 2|=3$（因为$G_1\cdot 2=\{2,4,5\}$）. Applying the theorem a third time gives $|(G_1)_2|=|(G_1)_2\cdot 3||((G_1)_2)_3|$. Any element of $G$ that fixes $1,2$ must send $3$ to either $3$ or $6$（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 $|(G_1)_2\cdot 3|=2$. One also sees that $((G_1)_2)_3$ consists only of the identity automorphism, as any element of $G$ fixing $1, 2$ and $3$ must also fix all other vertices, since they are determined by their adjacency to $1$, $2$ and $3$. Combining the preceding calculations, we can now obtain $|G| = 8 ⋅ 3 ⋅ 2 ⋅ 1 = 48$.

This result is mainly of use when $G$ and $X$ 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.

证明:
设$X=\{a_1,...,a_n\},G=\{g_1,...,g_m\}$，考虑一个表格，定义

$$e_{i,j}=\begin{cases} 1 & g_i(a_j)=a_j\\ 0 & g_i(a_j)\neq a_j \end{cases}.$$

我们考虑把每一行加起来，那么列表里元素和就是$\sum_{g\in G}|X^g|$。按列求和，那么元素和就是$\sum_{a\in X}|G_a|$。

我们知道$a\in X$ 可以根据轨道划分，

$$\sum_{a\in X}|G_a|=\sum_{G\cdot x\in X/G}\sum_{a\in G\cdot x}|G_a|.$$

我们知道：同一个 orbit 里的$a,b\in G\cdot x$，有$|G_a|=|G_b|$。因此根据 orbit-stabilizer 定理，

$$\sum_{G\cdot x\in X/G}\sum_{a\in G\cdot x}|G_a|=\sum_{G\cdot x\in X/G}|G\cdot x|\cdot |G_x|=|G|\cdot |X/G|.$$

因此

$$|G|\cdot |X/G|=\sum_{g\in G}|X^g|.$$

$\square$