---- > [!definition] Definition. ([[group action]]) >An **action** a [[group]] $G$ on an object $A$ of a [[category]] $\mathsf{C}$ is a [[group homomorphism|homomorphism]] $\sigma: G \to \text{Aut}_{\mathsf{C}}(A),$ >where $\text{Aut}_{\mathsf{C}}(A)$ is the [[automorphism|automorphism group]] of $A$ in $\mathsf{C}$. > >The case $\mathsf{C}= \mathsf{Set}$ is most important[^1], and indeed the default. A description of $\sigma$ in this context can be found in [[homomorphism induced by group action]] and/or [[permutation representation]]. More standard is the following definition, whose data is equivalent. > > Let $G$ be a [[group]] and $X$ a set. A **(left) group action** is a map $\begin{align} & G \times X \to X \\ & (g,x) \mapsto g \cdot x \end{align}$ satisfying >1. $e_{G}\cdot x = x$ for all $x \in X$; >2. $h \cdot (g \cdot x) = (hg)\cdot x$ for all $g,h \in G$ and all $x \in X$. > > We say the set $X$ **carries the action**. ^7ac4f4 > [!equivalence] Equivalence. (Group actions as functors) > [[monoid|Recalling]] how a [[group]] $G$ may be seen as a one-object [[category]] with the elements of $G$ as morphisms and composition via the group operation, a [[covariant functor|functor]] $\mathscr{F}:G \to \mathsf{Set}$ is precisely a [[group action]]. Indeed, to specify $\mathscr{F}$ one selects a set $X=\mathscr{F}(G)$ and a function $\text{Hom}(G, G) \to \text{Hom}(X,X);$ > (note that $\text{Hom}_{G}(G,G)$ in this one-object-categorical case is just $G$), every element $g \in \text{Hom}(G,G)$ is an [[isomorphism]], and functoriality ensures (1) that image $\mathscr{F}g$ is too, (2) that the map is a [[group homomorphism]]. In other words, $\mathscr{F}$ in fact corresponds to a [[group homomorphism|homomorphism]] $G \to \text{Aut}(X)$ which is precisely the [[group action]] definition. ^equivalence [^1]: Coming in second is $\mathsf{C}=k$-$\mathsf{Vect}$, in which case $\sigma$ is a [[group representation]]. > [!basicexample] Example. ([[permutation|Permutations]]) > Let $G$ be the [[symmetric group|group of permutations]] of $\{ 1,2,3,4 \}$ and $X:=\{ 1,2,3,4 \}$. Then $G$ acts on $X$ via the map $(\sigma, x) \mapsto \sigma(x)$ > for a [[permutation]] $\sigma$ in $G$ and element $x \in \{ 1,2,3,4 \}$. Importantly, $G$ also acts on the [[power set]] of $\{ 1,2,3,4 \}$ via the map $(\sigma, S) \mapsto \{ (\sigma(s): s \in S) \}$ > where $g \in G$ and $S \subset \{ 1,2,3,4 \}$. ^e828b3 > [!basicexample] Example. ($n$-gon Symmetries) > The [[dihedral group]] $D_{n}$ acts 'naturally' on the vertices of the regular $n$-gon. ![[CleanShot 2023-09-21 at [email protected]|200]]. In light of [[dihedral group and linear isomorphisms of the plane]], this action feels '[[linear map|linear]]' in some sense— this leads to representation theory. ^f2df52 > [!basicexample] Example. ([[conjugate|Conjugation]]) > Take $X=G$. Then $G$ acts on itself via the **left translation** map $\begin{align} G \times G & \to G \\ (g,s) & \to gs. \end{align}$ But be careful! You might be tempted to right-multiply, defining $\textcolor{Apricot}{(g,s) \to sg}.$ This does not work because since the definition requires $h (g \cdot s)=(hg)\cdot s$, we would have $hsg=shg$ which is not true in general. What *does* work is $\begin{align} G \times G & \to G \\ (g,s) & \to sg^{-1}. \end{align}$ And putting together the two maps we get a third $\begin{align} G \times G & \to G \\ (g,s) & \to gsg^{-1}, \end{align}$ which takes $s$ to the its [[conjugate|conjugation]] by $g$. So, $G$ can act on itself by 'changing the perspective' from which its elements are viewed. ^363027 ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```