note-math

Bijection forms a group

Mapping is associative

Composition of mappings is associative. You can compose functions from either the pre or post: or

for octonion with non associative multiplication, for the action defined by multiplication octonion, its composition cannot represented by multiplication

When a bijection acts on a certain structure of , there is a structure group that preserves the structure, which is a subgroup of

Example preserves linear structure

let be a subgroup of . let

[group-action]

[orbit] :=

Example acts on , orbit

[isotropy] :=

Example acts on , isotropy = rotation around the axis where is located, which is an embedded

is a subgroup of . a map that fix a point constitutes a subgroup of , is the intersection of the action group of and this fix mapping subgroup

Isotropy after changing the orbit base point

Mapping

• Homomorphism
• Bijection

[isotropy-in-same-orbit-is-isom] The isotropy of , , is written as , which is isomorphic to

According to the inverse image of acting on , decompose into the subgroup and its coset

Calculate the inverse image of ,

[orbit-istropy-theorem] There exists a bijection

Therefore,

set of cosets is isomorphic to the orbit . so which

Example let be a finite group, let . is a finite set and is a subgroup. There exists a smallest such that , thus . Let the group act on the coset space , isotropy , then or is divisible by

Change the base point of the orbit. forall ==>

Proof

[decomposition-into-orbit] Proof

Example , different orbits are spheres of different radii

The set of orbits :=

[Burnside-theorem] …

[conjugate-action] Conjugate action

as the transformation of the coordinates of the action of caused by changing the coordinates for any acted space

Example Representations of linear maps in different coordinates. Representations of manifold maps in different coordinates

The orbit of the conjugate action is called [conjugate-class]

Example The conjugate-class of a permutation is a cycle

Commutator [commutator]

[action-surjective] alias [action-transitive] := The following definitions are equivalent

• is a surjective

Example acting on is not transitive. acting on is transitive

[action-injective] alias [action-free] := The following definitions are equivalent

• Each orbit is a copy of
• is an injective