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 is a group with identity element , and is a set, then a (left) group action of on is a function
that satisfying the following two axioms:
- Identity: ;
- Compatibility: .
where is often shortened to .
The action of on is called transitive if for any two points , there exists a such that .(联通性)
Consider a group acting on a set . The orbit of an element is the set of element in to which can be moved by the elements of .
The orbit of is denoted by :
实际上 group action 诱导出了一个 上的等价关系: 当且仅当,也就是说 是联通的。
因此,orbits 就构成了 的一个划分:要么,要么。
而且不难发现,group action 是 transitive 的当且仅当只有一个 orbit:。
The set of all orbits of under the action of is written as , 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 , by contrast with the invariants (fixed points), denoted : the coinvariants are a quotient while the invariants are a subset.
A subset is said to be invariant under if , which is equivalent to .
Moreover, is said to be fixed under if for all .
Every orbit is an invariant subset of on which acts transitively. Conversely, any invariant subset of is a union of orbits.
Given with , it is said that is a fixed point of or fixes .
For every , the stabilizer subgroup of with respect to (also called the isotropy group or little group) is the set of all elements in that fixes :
This is a subgroup of , though typically not a normal one.
The action of on is free if and only if all stabilizers are trivial.
The kernel of the homomorphism with the symmetric group, , is given by the intersection of the stabilizers for all . If is trivial, the action is said to be faithful (or effective).
Let such that for some , then the two stabilizer groups are related by .
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 (that is, the set of all conjugates of the subgroup). Let denote the conjugacy class of . Then the orbit has type if the stabilizer of some/any in belongs to . A maximal orbit type is often called a principal orbit type.
(Orbit-stabilizer Theorem)
Orbits and stabilizers are closely related. For a fixed in , consider the map given by . By definition the image of this map is the orbit . The condition for two elements to have the same image is
In other words, if and only if and lie in the same coset for the stabilizer subgroup .
Thus, the fiber of over any in is contained in such a coset, and every such coset also occurs as a fiber.
Therefore induces a bijection between the set of cosets for the stabilizer subgroup and the orbit , which sends . This result is known as the orbit-stabilizer theorem.
If is finite then the orbit-stabilizer theorem, together with Lagrange’s theorem, gives
Example: Let be a group of prime order acting on a set with elements.
Since each orbit has either or elements(因为素数阶群必然是循环群), there are at least orbits of length (因为 orbits 构成 的一个划分,而且 orbits 大小要么是 要么是,所以至少有 个 orbits 大小为 1), which are -invariant elements. More specifically, and the number of -invariant elements are congruent modulo .
这个结果特别有用,因为它可以用于计数论证(通常在 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 denote its automorphism group. Then acts on the set of vertices , and this action is transitive as can be seen by composing rotations about the center of the cube. Thus, by the orbit-stabilizer theorem, (因为 action 是 transitive 的,所以). Applying the theorem now to the stabilizer , we can obtain . Any element of that fixes must send to either , , or . Thus, (因为). Applying the theorem a third time gives . Any element of that fixes must send to either or (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 . One also sees that consists only of the identity automorphism, as any element of fixing and must also fix all other vertices, since they are determined by their adjacency to , and . Combining the preceding calculations, we can now obtain .
(Burnside’s lemma)
where is the set of elements fixed by .
This result is mainly of use when and 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.