---- > [!definition] Definition. ([[permutation representation]]) > A **permutation representation** of a [[group]] $G$ is a [[group representation|representation]] of $G$ by the [[symmetric group]], i.e., is a [[group homomorphism|homomorphism]] $\varphi: G \to S_{n}.$ > \ > There is a bijective correspondence $\begin{bmatrix}\text{actions of }G \\ \text{on S}:= [n] \end{bmatrix}\leftrightarrow\begin{bmatrix} \text{permutation } \\ \text{representations } G \to S_{n} \end{bmatrix} . $ > > > Given a [[subgroup]] $H \leq G$, the [[Todd-Coxeter algorithm]] yields the **permutation representation** of $G$ corresponding to its [[group action|action]] on the right [[coset|cosets]] of $H$. > [!justification] > The proof of the correspondence is simple. Take an action $G \times S \to S, \ (g,s) \mapsto g \cdot s$. For each $g \in G$ define $g_{(\cdot)}: [n] \to [n] \in S_{n}$ to be given by $g_{(\cdot)}(j):=g \cdot j$. Then we have a [[group homomorphism]] $\begin{align} \varphi:G \to S_{n} \\ \varphi(g) := g_{(\cdot)}, \end{align}$ for given any $j \in [n]$, $\varphi(gh)(j)=(gh)_{(\cdot)}(j)=(gh)\cdot j=g\cdot(h\cdot j)=g_{(\cdot)}(h \cdot j)=g_{(\cdot)}h_{(\cdot)}(j)=\varphi(g)\varphi(h)(j)$. \ A sort of corollary is found in [[homomorphism induced by group action]]. ---- #### ---- #### 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 > ```