note-math

Bijection $𝑓:𝑋\leftrightarrow 𝑋$ forms a group

Mapping is associative

Composition of mappings is associative. You can compose functions from either the pre or post: $𝑓⇝𝑔\circ 𝑓$ or $𝑓⇝𝑓\circ 𝑔$

for octonion with non associative multiplication, for the action defined by multiplication octonion, its composition cannot represented by multiplication $𝑎\circ 𝑏\ne 𝑎\cdot 𝑏$

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

Example $\text{GL}$ preserves #link(<linear>)[linear structure]

let $𝐺$ be a subgroup of $𝑋!$. let $𝑥\in 𝑋$

group-action_(tag)

$$\begin{array}{ccc}𝐺\times 𝑋& ⟶& 𝑋\\ \left(𝑔,𝑥\right)& ⟿& 𝑔\cdot 𝑥\end{array}$$

orbit_(tag) :=

$$𝐺\cdot 𝑥=\left\{𝑔𝑥\in 𝑋:𝑔\in 𝐺\right\}=\text{ im }\left\{\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}\right\}$$

Example $\text{SO}\left(3\right)$ acts on ${\mathrm{ℝ}}^3$, orbit ${\mathrm{𝕊}}^2\left(\left|𝑥\right|\right)$

isotropy_(tag) :=

$${𝐺}_{𝑥}=\left\{𝑔\in 𝐺:𝑔𝑥=𝑥\right\}={\text{ im }}^{-1}\left\{\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}\right\}\left(𝑥\right)$$

Example $\text{SO}\left(3\right)$ acts on ${\mathrm{ℝ}}^3$, isotropy = rotation around the axis where $𝑥\in {\mathrm{𝕊}}^2$ is located, which is an embedded $\text{SO}\left(2\right)$

${𝐺}_{𝑥}$ is a subgroup of $𝐺$. a map $𝑓$ that fix a point $𝑥\in 𝑋$ constitutes a subgroup of $𝑋!$, ${𝐺}_{𝑥}$ is the intersection of the action group of $𝐺$ and this fix $𝑥$ mapping subgroup

Isotropy after changing the orbit base point $𝑥⇝ℎ𝑥$

$$\begin{array}{ccc}𝑔\left(ℎ𝑥\right)=ℎ𝑥& ⟺& {ℎ}^{-1}𝑔ℎ𝑥=𝑥\\ & ⟺& {ℎ}^{-1}𝑔ℎ\in {𝐺}_{𝑥}\end{array}$$

Mapping $\begin{array}{ccc}𝐺& ⟶& 𝐺\\ 𝑔& ⟿& {ℎ}^{-1}𝑔ℎ\end{array}$

• Homomorphism ${ℎ}^{-1}\left(𝑔\cdot {𝑔}^{\prime }\right)ℎ=\left({ℎ}^{-1}𝑔ℎ\right)\cdot \left({ℎ}^{-1}{𝑔}^{\prime }ℎ\right)$
• Bijection ${ℎ}^{-1}𝑔ℎ={𝑔}^{\prime }⟺𝑔=ℎ{𝑔}^{\prime }{ℎ}^{-1}$

isotropy-in-same-orbit-is-isom_(tag) The isotropy of $ℎ𝑥$, ${𝐺}_{ℎ𝑥}$, is written as $ℎ{𝐺}_{𝑥}{ℎ}^{-1}$, which is isomorphic to ${𝐺}_{𝑥}$

According to the inverse image of $𝐺$ acting on $𝐺𝑥$, decompose $𝐺$ into the subgroup ${𝐺}_{𝑥}$ and its coset $ℎ{𝐺}_{𝑥}$

$𝐺={⨆}_{𝑦\in 𝐺𝑥}{\text{ im }}^{-1}\left\{\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}\right\}\left(𝑦\right)$

Calculate the inverse image of $𝑦=ℎ𝑥\in 𝐺𝑥$, $𝑔𝑥=ℎ𝑥⟺{ℎ}^{-1}𝑔\in {𝐺}_{𝑥}⟺𝑔\in ℎ{𝐺}_{𝑥}$

$\left|{\text{im }}^{-1}\left\{\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}\right\}\left(𝑦\right)\right|=|ℎ{𝐺}_{𝑥}|=\left|{𝐺}_{𝑥}\right|$

orbit-istropy-theorem_(tag) There exists a bijection

$$\begin{array}{ccc}𝐺𝑥\times {𝐺}_{𝑥}& ⟷& 𝐺=\underset{𝑦\in 𝐺𝑥}{⨆}\cdots \\ \left(𝑦,\cdots \right)& ⟿& \cdots \end{array}$$

Therefore, $\left|𝐺\right|=|𝐺𝑥|\cdot \left|{𝐺}_{𝑥}\right|$

set of cosets is isomorphic to the orbit $\frac{𝐺}{{𝐺}_{𝑥}}\simeq 𝐺𝑥$. so $|𝐺𝑥|=\frac{\left|𝐺\right|}{\left|{𝐺}_{𝑥}\right|}$ which $\le \left|𝐺\right|$

Example let $𝐺$ be a finite group, let $𝑎\in 𝐺$. $𝐻=\left\{{𝑎}_1,{𝑎}_2,\dots \right\}$ is a finite set and is a subgroup. There exists a smallest $𝑘\in \mathrm{\mathbb{N}}$ such that ${𝑎}^{𝑘}=\mathrm{𝟙}$, thus ${𝑎}^{-1}={𝑎}^{𝑘-1}$. Let the group $𝐺$ act on the coset space $\left\{𝑔𝐻:𝑔\in 𝐺\right\}$, isotropy ${𝐺}_{𝐻}=𝐻$, then $\frac{\left|𝐺\right|}{\left|𝐻\right|}=\frac{\left|𝐺\right|}{𝑘}\in \mathrm{\mathbb{N}}$ or $\left|𝐺\right|$ is divisible by $𝑘$

Change the base point of the orbit. forall $𝑦=ℎ𝑥$ ==> $𝐺𝑥=𝐺𝑦$

Proof

decomposition-into-orbit_(tag) $𝐺𝑥\ne 𝐺{𝑥}^{\prime }⟺𝐺𝑥\cap 𝐺{𝑥}^{\prime }=\varnothing$ Proof

Example $\text{SO}\left(3\right),{\mathrm{ℝ}}^3$, different orbits are spheres of different radii

The set of orbits :=

$$\frac{𝑋}{𝐺}≔\left\{𝐺𝑥\in \text{ Subset}\left(𝑋\right):𝑥\in 𝑋\right\}$$

Burnside-theorem_(tag) …

conjugate-action_(tag) Conjugate action

$${𝑐}_{ℎ}:\begin{array}{ccc}𝐺& ⟶& 𝐺\\ 𝑔& ⟿& ℎ𝑔{ℎ}^{-1}\end{array}$$

as the transformation of the coordinates of the action of $𝑔$ caused by changing the coordinates $ℎ$ for any acted space $𝑋$

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_(tag)

Example The conjugate-class of a permutation is a cycle

Commutator commutator_(tag)

$$\begin{array}{ccc}\left(ℎ𝑔{ℎ}^{-1}=𝑔\right)& ⟺& \left(ℎ𝑔⟺𝑔ℎ\right)\\ & ⟺& \mathrm{𝟙}={ℎ}^{-1}\cdot 𝑔\cdot ℎ\cdot {𝑔}^{-1}\end{array}$$

action-surjective_(tag) alias action-transitive_(tag) := The following definitions are equivalent

• $\left|\frac{𝑋}{𝐺}\right|=1$
• $\exists 𝑥\in 𝑋,𝐺𝑥=𝑋$
• $\forall 𝑥\in 𝑋,𝐺𝑥=𝑋$
• $\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}$ is a surjective $𝐺\twoheadrightarrow 𝑋$

Example $\text{SO}\left(3\right)$ acting on ${\mathrm{ℝ}}^3\setminus 0$ is not transitive. $\text{GL}\left(3,\mathrm{ℝ}\right)$ acting on ${\mathrm{ℝ}}^3\setminus 0$ is transitive

action-injective_(tag) alias action-free_(tag) := The following definitions are equivalent

• Each orbit is a copy of $𝐺$
• $𝑔𝑥=ℎ𝑥⟹𝑔=ℎ$
• $𝑔𝑥=𝑥⟹𝑔=\mathrm{𝟙}$
• $\begin{array}{ccc}𝐺& ⟶& 𝑋\\ 𝑔& ⟿& 𝑔𝑥\end{array}$ is an injective $𝐺\hookrightarrow 𝑋$