From f5b0e95efa505c2bce6915250ab3dfca6fca5be8 Mon Sep 17 00:00:00 2001 From: Derived Cat Date: Sat, 24 Aug 2024 04:29:49 -0400 Subject: [PATCH] Update group.tex --- group.tex | 592 ++++++++++++++++++++++++++++-------------------------- 1 file changed, 309 insertions(+), 283 deletions(-) diff --git a/group.tex b/group.tex index 456dd3e..765b3b9 100644 --- a/group.tex +++ b/group.tex @@ -101,16 +101,16 @@ \section{Group Homomorphism} \end{definition} \begin{theorem}{Fundamental Theorem on Homomorphisms}{} - Let $G,H$ be groups and $\varphi:G\to H$ be a group homomorphism. Define natural projection + Let $G,H$ be groups and $\varphi:G\to H$ be a group homomorphism. Define natural projection \begin{align*} - \pi:G&\longrightarrow G/\ker\varphi\\ - g&\longmapsto g\ker\varphi + \pi:G & \longrightarrow G/\ker\varphi \\ + g & \longmapsto g\ker\varphi \end{align*} Then there exists a unique group homomorphism $\overline{\varphi}:G/\ker\varphi\to H$ such that the following diagram commutes \begin{center} \begin{tikzcd}[ampersand replacement=\&] G \arrow[r, "\varphi"] \arrow[d, "\pi"'] \& H \\[0.3cm] - G/\ker\varphi \arrow[ru, dashed, "\exists!\,\overline{\varphi}"'] \& + G/\ker\varphi \arrow[ru, dashed, "\exists!\,\overline{\varphi}"'] \& \end{tikzcd} \end{center} Moreover, $\overline{\varphi}$ is injective and we have $ G/\ker\varphi\cong \mathrm{im}\varphi$. @@ -127,16 +127,16 @@ \section{Group Homomorphism} \begin{center} \begin{tikzcd}[ampersand replacement=\&] G \arrow[r, "\varphi"] \arrow[d, "\pi"'] \& H \\[0.3cm] - G/N \arrow[ru, dashed, "\exists !\,\widetilde{\varphi}"'] \& + G/N \arrow[ru, dashed, "\exists !\,\widetilde{\varphi}"'] \& \end{tikzcd} \end{center} \end{proposition} \proof{ - Since $N\subseteq \ker\varphi$, there is a canonical projection + Since $N\subseteq \ker\varphi$, there is a canonical projection \begin{align*} - p:G/N &\longrightarrow G/\ker\varphi\\ - gN &\longmapsto g\ker\varphi + p:G/N & \longrightarrow G/\ker\varphi \\ + gN & \longmapsto g\ker\varphi \end{align*} According to the following diagram, we can define $\widetilde{\varphi}$ by $\widetilde{\varphi}=\overline{\varphi}\circ p$. \begin{center} @@ -168,7 +168,7 @@ \subsection{Free Object} \begin{align*} \overline{s^{-1}s}\approx \overline{1},\quad \overline{ss^{-1}}\approx \overline{1}, \quad \overline{1s}\approx s,\quad \overline{s1}\approx \overline{1} \end{align*} - Define $\sim$ to be the equivalence relation generated by $\approx$. Let $\pi:W(S)\to W(S)/\sim, w\mapsto [w]_\sim$ denote the quotient map. It is easy to see that for any $[w]_\sim\in W(S)/\sim$, there exists a unique representative element $\rho(w)\in W(S)$ which has shortest length among all representatives of $[w]_\sim$. A word $w$ is called \textbf{reduced} if $\rho(w)=w$. + Define $\sim$ to be the equivalence relation generated by $\approx$. Let $\pi:W(S)\to W(S)/\sim, w\mapsto [w]_\sim$ denote the quotient map. It is easy to see that for any $[w]_\sim\in W(S)/\sim$, there exists a unique representative element $\rho(w)\in W(S)$ which has shortest length among all representatives of $[w]_\sim$. A word $w$ is called \textbf{reduced} if $\rho(w)=w$. \end{definition} @@ -177,12 +177,12 @@ \subsection{Free Object} \begin{center} \begin{tikzcd}[ampersand replacement=\&] \mathrm{Free}_{\mathsf{Grp}}(S)\arrow[r, dashed, "\exists !\,\widetilde{f}"] \& G \\[0.3cm] - S\arrow[u, "\iota"] \arrow[ru, "f"'] \& + S\arrow[u, "\iota"] \arrow[ru, "f"'] \& \end{tikzcd} \end{center} The free group $\mathrm{Free}_{\mathsf{Grp}}(S)$ can be contructed as follows: as a set it consists of all reduced words on $S$. The binary operation $\cdot$ is concatenation with reduction defined by \[ - w_1\cdot w_2 = \rho(w_1\diamond w_2). + w_1\cdot w_2 = \rho(w_1\diamond w_2). \] The identity element is the empty word. The inverse of a word is obtained by reversing the order of the letters and replacing each letter by its inverse. \end{definition} @@ -192,22 +192,22 @@ \subsection{Free Object} The forgetful functor $U:\mathsf{Grp}\to \mathsf{Set}$ forgets the group structure of a group and returns the underlying set. \begin{enumerate}[(i)] \item $U$ is representable by $\left(\mathbb{Z},1\right)$. The natural isomorphism $\phi:\mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},-\right)\xRightarrow{\sim} U$ is given by - \begin{align*} - \phi_G:\mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},G\right)&\xlongrightarrow{\sim} U(G)\\ - f&\longmapsto f(1). - \end{align*} - A group homomorphism from $\mathbb{Z}$ to $G$ is uniquely determined by its action on $1$. - \item $U$ is faithful but not full. + \begin{align*} + \phi_G:\mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},G\right) & \xlongrightarrow{\sim} U(G) \\ + f & \longmapsto f(1). + \end{align*} + A group homomorphism from $\mathbb{Z}$ to $G$ is uniquely determined by its action on $1$. + \item $U$ is faithful but not full. \end{enumerate} \end{example} \begin{prf} \begin{enumerate}[(i)] - \item $\phi:\mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},-\right)\xRightarrow{\sim} U$ is the composition of the following natural isomorphisms - \[ - \mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},-\right)\cong\mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(\{*\}),-)\cong \mathrm{Hom}_{\mathsf{Set}}(\{*\},U(-))\cong U. - \] - \item $U$ is not full because not every mapping $f:\mathbb{Z}\to G$ is a group homomorphism. + \item $\phi:\mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},-\right)\xRightarrow{\sim} U$ is the composition of the following natural isomorphisms + \[ + \mathrm{Hom}_{\mathsf{Grp}}\left(\mathbb{Z},-\right)\cong\mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(\{*\}),-)\cong \mathrm{Hom}_{\mathsf{Set}}(\{*\},U(-))\cong U. + \] + \item $U$ is not full because not every mapping $f:\mathbb{Z}\to G$ is a group homomorphism. \end{enumerate} \end{prf} @@ -215,24 +215,24 @@ \subsection{Free Object} \begin{proposition}{Free-Forgetful Adjunction $\mathrm{Free}_{\mathsf{Grp}}\dashv U$}{} The free group functor $\mathrm{Free}_{\mathsf{Grp}}$ is left adjoint to the forgetful functor $U:\mathsf{Grp}\to \mathsf{Set}$ $$ - \begin{tikzcd}[ampersand replacement=\&] - \mathsf{Set} \arrow[rr, "\mathrm{Free}_{\mathsf{Grp}}", bend left] \&[-10pt]\bot\&[-10pt] \mathsf{Grp} \arrow[ll, "U", bend left] + \begin{tikzcd}[ampersand replacement=\&] + \mathsf{Set} \arrow[rr, "\mathrm{Free}_{\mathsf{Grp}}", bend left] \&[-10pt]\bot\&[-10pt] \mathsf{Grp} \arrow[ll, "U", bend left] \end{tikzcd} - $$ + $$ The adjunction isomorphism is given by \begin{align*} - \varphi_{S,G}:\mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(S),G)&\xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{Set}}(S,U(G))\\ - g&\longmapsto g\circ \iota + \varphi_{S,G}:\mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(S),G) & \xlongrightarrow{\sim} \mathrm{Hom}_{\mathsf{Set}}(S,U(G)) \\ + g & \longmapsto g\circ \iota \end{align*} \end{proposition} \begin{prf} First we show that $\varphi_{S,G}$ is injective. Suppose $g_1,g_2:\mathrm{Free}_{\mathsf{Grp}}(S)\to G$ are two group homomorphisms such that $g_1\circ \iota=g_2\circ \iota$. By the universal property of free group, we have $g_1=g_2$. Then we show that $\varphi_{S,G}$ is surjective. Suppose $f:S\to U(G)$ is a function. By the universal property there exists a group homomorphism $\widetilde{f}:\mathrm{Free}_{\mathsf{Grp}}(S)\to G$ such that $\varphi_{S,G}(\widetilde{f})=\widetilde{f}\circ \iota=f$. Finally, we show that $\varphi_{S,G}$ is natural in $S$ and $G$. Suppose $h:S_1\to S_2$ is a function and $q:G_1\to G_2$ is a group homomorphism. Then we can check that for any $g\in \mathrm{Hom}_{\mathsf{Grp}}(\mathrm{Free}_{\mathsf{Grp}}(S_2),G_2)$, \begin{align*} - \varphi_{S_1,G_1}(q\circ g\circ \iota_{S_1})&=(q\circ g\circ \iota_{S_1})\circ \iota_{S_1}\\ - &=q\circ g\circ (\iota_{S_1}\circ \iota_{S_1})\\ - &=q\circ g\circ \iota_{S_2}\\ - &=\varphi_{S_2,G_2}(g\circ \iota_{S_2}). + \varphi_{S_1,G_1}(q\circ g\circ \iota_{S_1}) & =(q\circ g\circ \iota_{S_1})\circ \iota_{S_1} \\ + & =q\circ g\circ (\iota_{S_1}\circ \iota_{S_1}) \\ + & =q\circ g\circ \iota_{S_2} \\ + & =\varphi_{S_2,G_2}(g\circ \iota_{S_2}). \end{align*} \end{prf} @@ -257,7 +257,7 @@ \subsection{Inverse Limit} \begin{example}{Inverse Limit $\varprojlim_{i\ge 1}G_i$}{} Let $\mathsf{I}=\left(\mathbb{Z}_{\ge 1},\le\right)$ be a filtered thin category and $F:\mathsf{I}^{\mathrm{op}}\to \mathsf{Grp}$ be a functor. To determine an inverse system, it is sufficient to specify $G_i$ and $f_{i,i+1}:G_{i+1}\to G_i$ for all $i\in \mathbb{Z}_{\ge 1}$. The inverse limit of this inverse system is denoted by $\varprojlim_{i\ge 1}G_i$, which we now write as $G$ for simplicity.\\ - $G$ can be imaged as a tree with root layer being $G_0=\{1\}$ and $i$-th layer being $G_i$. Each node in $G_i$ has a unique parent node in $G_{i-1}$, which is determined by $f_{i,i-1}$. An element in $G$ is a path starting from the root and passing through each $G_i$ exactly once along the edges of the tree. The $i$-th component $x_i$ of an element $x\in G$ includes all information of its history path from $G_0$ to $G_i$, which makes $x_1,x_2,\cdots, x_{i-1}$ redundant. + $G$ can be imaged as a tree with root layer being $G_0=\{1\}$ and $i$-th layer being $G_i$. Each node in $G_i$ has a unique parent node in $G_{i-1}$, which is determined by $f_{i,i-1}$. An element in $G$ is a path starting from the root and passing through each $G_i$ exactly once along the edges of the tree. The $i$-th component $x_i$ of an element $x\in G$ includes all information of its history path from $G_0$ to $G_i$, which makes $x_1,x_2,\cdots, x_{i-1}$ redundant. \end{example} @@ -270,22 +270,22 @@ \subsection{Definitions} \begin{definition}{Group Action}{} Let $G$ be a group and $X$ be a set. A \textbf{group action} of $G$ on $X$ is a group homomorphism \begin{align*} - \sigma:G&\longrightarrow \mathrm{Aut}_{\mathsf{Set}}(X)\\ - g&\longmapsto \sigma_g + \sigma:G & \longrightarrow \mathrm{Aut}_{\mathsf{Set}}(X) \\ + g & \longmapsto \sigma_g \end{align*} If $G$ acts on $X$ by $\sigma$, we say $(X,\sigma)$ is a \textbf{$G$-set}. If there is no ambiguity, we simply say $X$ is a $G$-set. \end{definition} The $G$-sets and $G$-maps form a category $G\text{-}\mathsf{Set}$ and we have category isomorphism \begin{align*} - G\text{-}\mathsf{Set}&\stackrel{\sim}{\longrightarrow}[\mathsf{B}G, \mathsf{Set}] \\ - \sigma&\longmapsto \sigma(-) + G\text{-}\mathsf{Set} & \stackrel{\sim}{\longrightarrow}[\mathsf{B}G, \mathsf{Set}] \\ + \sigma & \longmapsto \sigma(-) \end{align*} \begin{proposition}{Equivalent Definition of Group Actions}{} Let $G$ be a group and $X$ be a set. A group action of $G$ on $X$ can be alternatively defined as a map \begin{align*} - \cdot:G\times X&\longrightarrow X\\ - (g,x)&\longmapsto g\cdot x + \cdot:G\times X & \longrightarrow X \\ + (g,x) & \longmapsto g\cdot x \end{align*} such that \begin{enumerate}[(i)] @@ -294,7 +294,7 @@ \subsection{Definitions} \end{enumerate} The equivalence of the two definitions is given by \[ - \sigma_g(x)=g\cdot x. + \sigma_g(x)=g\cdot x. \] \end{proposition} @@ -306,30 +306,30 @@ \subsection{Definitions} \begin{example}{Actions on $X$ Induce Actions on $2^X$}{acting_on_power_set} If a group $G$ acts on a set $X$, then $G$ acts on the power set $2^X$ by \[ - g\cdot A=\{ g\cdot x\mid x\in A\} . + g\cdot A=\{ g\cdot x\mid x\in A\} . \] \end{example} \begin{definition}{Equivariant Map}{} Let $G$ be a group and $X,Y$ be $G$-sets. A map $f:X\to Y$ is called \textbf{equivariant} if for all $g\in G$ and $x\in X$, we have \[ - f(g\cdot x)=g\cdot f(x) . + f(g\cdot x)=g\cdot f(x) . \] Equivalently, $f$ is equivariant if it is a natural transformation $f:\sigma(-)\implies \sigma'(-)$ \[ \begin{tikzcd}[ampersand replacement=\&, column sep=1.7em, row sep=small] \mathsf{B}G \& \bullet \arrow[rr, "g"] \& \& \bullet \\ - \& X \arrow[dd, "f"'] \arrow[rr, "\sigma_g"] \& \& X \arrow[dd, "f"] \\ + \& X \arrow[dd, "f"'] \arrow[rr, "\sigma_g"] \& \& X \arrow[dd, "f"] \\ \mathsf{Set} \& \& \& \\ - \& Y \arrow[rr, "\sigma'_g"'] \& \& Y - \end{tikzcd} + \& Y \arrow[rr, "\sigma'_g"'] \& \& Y + \end{tikzcd} \] \end{definition} \begin{definition}{Product of $G$-Sets}{} The \textbf{product} of two $G$-sets $X$ and $Y$ is defined as the set $X\times Y$ with the $G$-action \[ - g\cdot (x,y)=(g\cdot x, g\cdot y) . + g\cdot (x,y)=(g\cdot x, g\cdot y) . \] Alternatively, the product of two $G$-sets can be defined as the product of two functors, cf. \Cref{th:ev_functor_preserves_limits}. \end{definition} @@ -337,10 +337,10 @@ \subsection{Definitions} \begin{definition}{Coproduct of $G$-Sets}{} The \textbf{coproduct} of two $G$-sets $X$ and $Y$ is defined as the set $X\sqcup Y$ with the $G$-action \[ - g\cdot a=\begin{cases} - g\cdot a & a\in X\\ - g\cdot a & a\in Y - \end{cases} + g\cdot a=\begin{cases} + g\cdot a & a\in X \\ + g\cdot a & a\in Y + \end{cases} \] Alternatively, the coproduct of two $G$-sets can be defined as the coproduct of two functors, cf. \Cref{th:ev_functor_preserves_limits}. \end{definition} @@ -348,37 +348,37 @@ \subsection{Definitions} \begin{example}{$\mathrm{Aut}_\mathsf{C}(X)$ acts on $\mathrm{Hom}_\mathsf{C}(X,Y)$ and $\mathrm{Hom}_\mathsf{C}(Y,X)$}{aut_acts_on_hom} Let $X$ and $Y$ be objects in a category $\mathsf{C}$. Then $\mathrm{Aut}_\mathsf{C}(X)$ acts on $\mathrm{Hom}_\mathsf{C}(X,Y)$ by the composition of functors \[ - \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] - \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r,"{(-)}^{-1}"] \&[-3em] \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)^{\mathrm{op}} \arrow[r, hook] \&[-2em] \mathsf{C}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(-,Y)}"] \& \mathsf{Set}\\[-2.3em] - \bullet \arrow[r, maps to] \arrow[d, "g"'] \& \bullet \arrow[r, maps to] \arrow[d, "g^{-1}"] \& X \arrow[d, "g^{-1}"] \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} \arrow[d, "\left(g^{-1}\right)^*"] \\ - \bullet \arrow[r, maps to] \& \bullet \arrow[r, maps to] \& X \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} - \end{tikzcd} + \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] + \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r,"{(-)}^{-1}"] \&[-3em] \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)^{\mathrm{op}} \arrow[r, hook] \&[-2em] \mathsf{C}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(-,Y)}"] \& \mathsf{Set}\\[-2.3em] + \bullet \arrow[r, maps to] \arrow[d, "g"'] \& \bullet \arrow[r, maps to] \arrow[d, "g^{-1}"] \& X \arrow[d, "g^{-1}"] \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} \arrow[d, "\left(g^{-1}\right)^*"] \\ + \bullet \arrow[r, maps to] \& \bullet \arrow[r, maps to] \& X \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} + \end{tikzcd} \] Writing explicily, the action is given by \begin{align*} - \mathrm{Aut}_\mathsf{C}(X)\times \mathrm{Hom}_\mathsf{C}(X,Y)&\longrightarrow \mathrm{Hom}_\mathsf{C}(X,Y)\\ - (g,f)&\longmapsto f\circ g^{-1} + \mathrm{Aut}_\mathsf{C}(X)\times \mathrm{Hom}_\mathsf{C}(X,Y) & \longrightarrow \mathrm{Hom}_\mathsf{C}(X,Y) \\ + (g,f) & \longmapsto f\circ g^{-1} \end{align*} Similarly, $\mathrm{Aut}_\mathsf{C}(Y)$ acts on $\mathrm{Hom}_\mathsf{C}(Y,X)$ by \[ - \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] - \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r, hook] \&[-1em] \mathsf{C} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(Y,-)}"] \& \mathsf{Set}\\[-2.3em] - \bullet \arrow[r, maps to] \arrow[d, "g"'] \& X \arrow[d, "g"] \arrow[r, maps to] \& {\mathrm{Hom}(Y,X)} \arrow[d, "g_*"] \\ - \bullet \arrow[r, maps to] \& X \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} - \end{tikzcd} + \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] + \mathsf{B}\mathrm{Aut}_\mathsf{C}(X)\arrow[r, hook] \&[-1em] \mathsf{C} \arrow[r, "{\mathrm{Hom}_{\mathsf{C}}(Y,-)}"] \& \mathsf{Set}\\[-2.3em] + \bullet \arrow[r, maps to] \arrow[d, "g"'] \& X \arrow[d, "g"] \arrow[r, maps to] \& {\mathrm{Hom}(Y,X)} \arrow[d, "g_*"] \\ + \bullet \arrow[r, maps to] \& X \arrow[r, maps to] \& {\mathrm{Hom}(X,Y)} + \end{tikzcd} \] Writing explicily, the action is given by \begin{align*} - \mathrm{Aut}_\mathsf{C}(Y)\times \mathrm{Hom}_\mathsf{C}(Y,X)&\longrightarrow \mathrm{Hom}_\mathsf{C}(Y,X)\\ - (g,f)&\longmapsto g\circ f + \mathrm{Aut}_\mathsf{C}(Y)\times \mathrm{Hom}_\mathsf{C}(Y,X) & \longrightarrow \mathrm{Hom}_\mathsf{C}(Y,X) \\ + (g,f) & \longmapsto g\circ f \end{align*} \end{example} \begin{example}{Actions on $X$ Induce Actions on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$}{acting_on_functions} - If $G$ acts on $X$ through a functor $\sigma(-):\mathsf{B}G\to\mathsf{Set}$, then it also acts on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ for any set $Y$ by the composition of functors + If $G$ acts on $X$ through a functor $\sigma(-):\mathsf{B}G\to\mathsf{Set}$, then it also acts on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ for any set $Y$ by the composition of functors \[ - \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] - \mathsf{B}G \arrow[r,"{(-)}^{-1}"] \&[-2em] \mathsf{B}G^{\mathrm{op}} \arrow[r, "\sigma(-)^{\mathrm{op}}"] \&[-2.2em] \mathsf{Set}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{Set}}(-,Y)}"] \& \mathsf{Set} + \begin{tikzcd}[ampersand replacement=\&, column sep=5em, row sep=3em] + \mathsf{B}G \arrow[r,"{(-)}^{-1}"] \&[-2em] \mathsf{B}G^{\mathrm{op}} \arrow[r, "\sigma(-)^{\mathrm{op}}"] \&[-2.2em] \mathsf{Set}^{\mathrm{op}} \arrow[r, "{\mathrm{Hom}_{\mathsf{Set}}(-,Y)}"] \& \mathsf{Set} \end{tikzcd} \] The left action on $\mathrm{Hom}_{\mathsf{Set}}(X,Y)$ is given explicitly as follows: for all $g\in G$, $f\in \mathrm{Hom}_{\mathsf{Set}}(X,Y)$ and $x\in X$, @@ -389,41 +389,41 @@ \subsection{Definitions} \[ (f\star g)(x)=f(g\cdot x). \] - - + + \end{example} \begin{prf} We can check that \begin{align*} - (g_1\cdot (g_2\cdot f))(x)&=\left(g_2\cdot f\right)\left(g_1^{-1}\cdot x\right)\\ - &=\left(g_2\cdot f\right)\left(g_1^{-1}\cdot x\right)\\ - &=f\left(g_2^{-1}\cdot\left(g_1^{-1}\cdot x\right)\right)\\ - &=f\left(\left(g_2^{-1} g_1^{-1}\right)\cdot x\right)\\ - &=\left(\left(g_1g_2\right)\cdot f\right)(x). + (g_1\cdot (g_2\cdot f))(x) & =\left(g_2\cdot f\right)\left(g_1^{-1}\cdot x\right) \\ + & =\left(g_2\cdot f\right)\left(g_1^{-1}\cdot x\right) \\ + & =f\left(g_2^{-1}\cdot\left(g_1^{-1}\cdot x\right)\right) \\ + & =f\left(\left(g_2^{-1} g_1^{-1}\right)\cdot x\right) \\ + & =\left(\left(g_1g_2\right)\cdot f\right)(x). \end{align*} and also check that \begin{align*} ((f\star g_1)\star g_2)(x) - &=\left(f\star g_1\right)\left(g_2\cdot x\right)\\ - &=f\left(g_1\cdot\left(g_2\cdot x\right)\right)\\ - &=f\left((g_1g_2)\cdot x\right)\\ - &=\left(f\star (g_1 g_2)\right)(x). + & =\left(f\star g_1\right)\left(g_2\cdot x\right) \\ + & =f\left(g_1\cdot\left(g_2\cdot x\right)\right) \\ + & =f\left((g_1g_2)\cdot x\right) \\ + & =\left(f\star (g_1 g_2)\right)(x). \end{align*} \end{prf} \begin{definition}{Orbit of a Group Action}{} - Let $G$ be a group acting on a set $X$. For $x\in X$, the \textbf{orbit} of $x$ is defined as + Let $G$ be a group acting on a set $X$. For $x\in X$, the \textbf{orbit} of $x$ is defined as \[ - G x=\{ g\cdot x\mid g\in G\} . + G x=\{ g\cdot x\mid g\in G\} . \] \end{definition} \begin{definition}{Orbit Space}{} - Let $G$ be a group acting on a set $X$. The \textbf{orbit space} of $G$ acting on $X$ is defined as + Let $G$ be a group acting on a set $X$. The \textbf{orbit space} of $G$ acting on $X$ is defined as \[ - G\backslash X=\{ Gx\mid x\in X\}. + G\backslash X=\{ Gx\mid x\in X\}. \] \end{definition} @@ -431,17 +431,17 @@ \subsection{Definitions} \begin{proposition}{Orbit Decomposition}{} Let $G$ be a group acting on a set $X$. We define a equivalence relation $\sim$ on $X$ by \[ - x\sim y \iff Gx=Gy. + x\sim y \iff Gx=Gy. \] Then the equivalence class of $x$ is exactly $Gx$. The quotient set $X/\sim$ is exactly the orbit space $G\backslash X$. And we have a partition of $X$ by the orbits of $G$ acting on $X$ \[ - X=\bigsqcup_{Gx\in G\backslash X}Gx. + X=\bigsqcup_{Gx\in G\backslash X}Gx. \] \end{proposition} \begin{prf} -We can check that the equivalence class of $x$ is $Gx$. If $y\sim x$, then $y\in Gy=Gx$. If $y\in Gx$, then $Gy\subseteq Gx$ and $x\in Gy$. Note $x\in Gy$ implies $Gx\subseteq Gy$. We have $Gx=Gy$, i.e. $x\sim y$. + We can check that the equivalence class of $x$ is $Gx$. If $y\sim x$, then $y\in Gy=Gx$. If $y\in Gx$, then $Gy\subseteq Gx$ and $x\in Gy$. Note $x\in Gy$ implies $Gx\subseteq Gy$. We have $Gx=Gy$, i.e. $x\sim y$. \end{prf} @@ -449,15 +449,15 @@ \subsection{Definitions} \begin{definition}{$G$-invariant element }{} Let $G$ be a group acting on a set $X$. An element $x\in X$ is called \textbf{$G$-invariant} if $Gx=\{ x\}$ or equivalently $|Gx|=1$. The set of all $G$-invariant elements is denoted by $X^G$ \[ - X^G=\{ x\in X\mid Gx=\{ x\} \} = \{ x\in X\mid \forall g\in G, g\cdot x=x \} . + X^G=\{ x\in X\mid Gx=\{ x\} \} = \{ x\in X\mid \forall g\in G, g\cdot x=x \} . \] \end{definition} \begin{definition}{Stabilizer Subgroup}{} - Let $G$ be a group acting on a set $X$. For $x\in X$, the \textbf{stabilizer subgroup} of of $G$ with respect to $x$ is defined as + Let $G$ be a group acting on a set $X$. For $x\in X$, the \textbf{stabilizer subgroup} of of $G$ with respect to $x$ is defined as \[ - \mathrm{Stab}_G(x)=\{ g\in G\mid g\cdot x=x\} . + \mathrm{Stab}_G(x)=\{ g\in G\mid g\cdot x=x\} . \] It is easy to see that $\mathrm{Stab}_G(x)$ is a subgroup of $G$. \end{definition} @@ -490,6 +490,7 @@ \subsection{Definitions} \end{enumerate} \end{definition} +It is clear that a free action is faithful, but the converse does not hold in general. \begin{definition}{Transitive Group Action}{} Let $G$ be a group acting on a set $X$. The action is called \textbf{transitive} if any of the following equivalent conditions holds @@ -500,12 +501,37 @@ \subsection{Definitions} If $G$ acts transitively on $X$, then $X$ is called a \textbf{homogeneous space} for $G$. \end{definition} -\begin{example}{$G$ Acts on Orbit $Gx$ Transitively}{} +The following proposition shows that we can understand a group action on a set $X$ by studying the group action on each $G$-orbit $Gx$ separately. + +\begin{proposition}{$G$ Acts on Orbit $Gx$ Transitively}{} Let $G$ be a group acting on a set $X$ and $x\in X$. Then $G$ acts on the orbit $Gx$ by left multiplication transitively. And we have a $G$-set isomorphism \[ - X\cong \bigsqcup_{Gx\in G\backslash X}Gx, + X\cong \bigsqcup_{Gx\in G\backslash X}Gx, \] which decomposes any $G$-set into coproduct of transitive $G$-sets. +\end{proposition} + +\begin{example}{The Orbit Decomposition of Subgroup Action}{} + Let $G$ be a group acting on a set $X$ with orbit decomposition + \[ + X=\bigsqcup_{i \in I} G x_i + \] + Suppose $H$ be a subgroup of $G$ and $G$ has right coset decomposition + \[ + G=\bigsqcup_{j \in J} Hg_j + \] + Then $H$ also acts on $X$ and each $G$-orbit is disjoint union of some $H$-orbits, which can written as + \[ + Gx_i =\bigsqcup_{k \in K} H s_{ik}. + \] + More concretely, $Gx_i$ is the union of the cosets $H (g_jx_i)\;(j \in J)$, + \[ + Gx_i =\bigcup_{j \in J} H g_j x_i. + \] + But $H (g_{j}x_i)$ may coincide with $H (g_{j'}x_i)$ for $j\ne j'$. We can duplicate $H (g_jx_i)\;(j \in J)$ by checking if there exists $h \in H$ such that $h g_j s_i=g_{j'} s_i$. Suppose $a \sim_H b$ iff $a$ and $b$ in the same $H$-orbit. Then we get + \[ + \{s_{ik}\mid k \in K\}=\{g_j x_i \mid j \in J\}/\sim_H. + \] \end{example} \begin{definition}{Regular Group Action}{} @@ -519,26 +545,26 @@ \subsection{Definitions} \subsection{Coset} \begin{example}{Left Multiplication Action}{} - Let $G$ be a group. The \textbf{left multiplication action} of $G$ on itself is defined as + Let $G$ be a group. The \textbf{left multiplication action} of $G$ on itself is defined as \begin{align*} - m^L:G &\longrightarrow \mathrm{Aut}(G)\\ - g &\longmapsto ( x\longmapsto gx) + m^L:G & \longrightarrow \mathrm{Aut}(G) \\ + g & \longmapsto ( x\longmapsto gx) \end{align*} \end{example} \begin{example}{Right Multiplication Action}{} - Let $G$ be a group. The \textbf{right multiplication action} of $G$ on itself is defined as + Let $G$ be a group. The \textbf{right multiplication action} of $G$ on itself is defined as \begin{align*} - m^R:G^\circ &\longrightarrow \mathrm{Aut}(G)\\ - g &\longmapsto ( x\longmapsto xg) + m^R:G^\circ & \longrightarrow \mathrm{Aut}(G) \\ + g & \longmapsto ( x\longmapsto xg) \end{align*} \end{example} \begin{definition}{Left Cosets}{} Let $G$ be a group and $H$ be a subgroup of $G$. $H^\circ$ can act on $G$ through $H^\circ\hookrightarrow G^\circ\stackrel{m^R}{\longrightarrow} \mathrm{Aut}(G)$, namely \begin{align*} - H^\circ &\longrightarrow \mathrm{Aut}(G)\\ - h &\longmapsto (g\longmapsto gh) + H^\circ & \longrightarrow \mathrm{Aut}(G) \\ + h & \longmapsto (g\longmapsto gh) \end{align*} The orbit of $g$ under $H^\circ$ is called the \textbf{left coset} of $H$ containing $g$, denoted by $gH$ \[ @@ -550,8 +576,8 @@ \subsection{Coset} \begin{example}{$G$ Acts on $G/H$ Transitively}{} Let $G$ be a group and $H$ be a subgroup of $G$. $G$ acts on $G/H$ through \begin{align*} - G &\longrightarrow \mathrm{Aut}(G/H)\\ - g &\longmapsto (xH\longmapsto gxH) + G & \longrightarrow \mathrm{Aut}(G/H) \\ + g & \longmapsto (xH\longmapsto gxH) \end{align*} For any $xH,yH\in G/H$, we have $yH=gxH$ for some $g=yx^{-1}\in G$. Thus $G$ acts on $G/H$ transitively. \end{example} @@ -559,8 +585,8 @@ \subsection{Coset} \begin{definition}{Right Cosets}{} Let $G$ be a group and $H$ be a subgroup of $G$. $H$ can act on $G$ through $H\hookrightarrow G\stackrel{m^L}{\longrightarrow} \mathrm{Aut}(G)$, namely \begin{align*} - H &\longrightarrow \mathrm{Aut}(G)\\ - h &\longmapsto (g\longmapsto hg) + H & \longrightarrow \mathrm{Aut}(G) \\ + h & \longmapsto (g\longmapsto hg) \end{align*} The orbit of $g$ under $H$ is called the \textbf{right coset} of $H$ containing $g$, denoted by $Hg$ \[ @@ -581,51 +607,51 @@ \subsection{Coset} \begin{proposition}{$G$-Set Isomorphism $G/\mathrm{Stab}_G(x)\cong Gx$}{iso_stab_orbit} Let $G$ be a group acting on a set $X$ and $x\in X$. Then the map \begin{align*} - F:G/\mathrm{Stab}_G(x) &\longrightarrow Gx\\ - g\hspace{1pt}\mathrm{Stab}_G(x) &\longmapsto g\cdot x + F:G/\mathrm{Stab}_G(x) & \longrightarrow Gx \\ + g\hspace{1pt}\mathrm{Stab}_G(x) & \longmapsto g\cdot x \end{align*} is a $G$-set isomorphism. \end{proposition} -\begin{prf} - The map is well-defined since for any $h\in \mathrm{Stab}_G(x)$, we have $(gh)\cdot x=g\cdot (h\cdot x)=g\cdot x$. The map is a $G$-set homomorphism since for any $g_1,g_2\in G$, +\begin{prf} + The map is well-defined since for any $h\in \mathrm{Stab}_G(x)$, we have $(gh)\cdot x=g\cdot (h\cdot x)=g\cdot x$. The map is a $G$-set homomorphism since for any $g_1,g_2\in G$, \begin{align*} F\left(g_1\cdot g_2\mathrm{Stab}_G(x)\right)=(g_1g_2)\cdot x=g_1\cdot(g_2\cdot x)=g_1\cdot F\left(g_2\mathrm{Stab}_G(x)\right). \end{align*} - The map is injective since for any $g,h\in G$, if $g\cdot x=h\cdot x$, then $h^{-1}g\in \mathrm{Stab}_G(x)$. Hence $g\mathrm{Stab}_G(x)=h\mathrm{Stab}_G(x)$. The map is surjective because for any $g\cdot x\in Gx$, we have $F\left(g\hspace{1pt}\mathrm{Stab}_G(x)\right)=g\cdot x$. + The map is injective since for any $g,h\in G$, if $g\cdot x=h\cdot x$, then $h^{-1}g\in \mathrm{Stab}_G(x)$. Hence $g\mathrm{Stab}_G(x)=h\mathrm{Stab}_G(x)$. The map is surjective because for any $g\cdot x\in Gx$, we have $F\left(g\hspace{1pt}\mathrm{Stab}_G(x)\right)=g\cdot x$. \end{prf} \begin{theorem}{Orbit-Stabilizer Theorem}{} Let $G$ be a group acting on a set $X$. For $x\in X$, we have \[ - |G|=|Gx|\cdot |\mathrm{Stab}_G(x)| . + |G|=|Gx|\cdot |\mathrm{Stab}_G(x)| . \] \end{theorem} -\begin{prf} +\begin{prf} According to \Cref{th:iso_stab_orbit}, we have \[ -\left|Gx\right|=\left|G/\mathrm{Stab}_G(x)\right|=|G|/\left|\mathrm{Stab}_G(x)\right|. + \left|Gx\right|=\left|G/\mathrm{Stab}_G(x)\right|=|G|/\left|\mathrm{Stab}_G(x)\right|. \] \end{prf} \begin{theorem}{Burnside's Lemma}{Burnside's_lemma} Let $G$ be a finite group acting on a finite set $X$. Then the number of orbits of $G$ on $X$ is equal to \[ - |G\backslash X|=\frac{1}{|G|}\sum_{g\in G}|X^g| , + |G\backslash X|=\frac{1}{|G|}\sum_{g\in G}|X^g| , \] where $X^g=\{x\in X\mid g\cdot x=x\}$ is the set of fixed points of $g$. \end{theorem} -\begin{prf} +\begin{prf} \begin{align*} - \sum_{g \in G}\left|X^g\right|&=|\{(g, x) \in G \times X \mid g \cdot x=x\}|\\ - &=\sum_{x \in X}\left|\mathrm{Stab}_G(x)\right|\\ - &=\sum_{x \in X}\frac{|G|}{\left|G x\right|} \quad \text{by Orbit-Stabilizer Theorem}\\ - &=|G| \sum_{G y \in G \backslash X}\; \sum_{x \in Gy}\frac{1}{\left|G x\right|}\\ - &=|G| \sum_{G y \in G \backslash X} \left|Gy\right|\frac{1}{\left|G y\right|}\\ - &=|G| \sum_{G y \in G \backslash X} 1\\ - &=|G| \cdot|G \backslash X|. + \sum_{g \in G}\left|X^g\right| & =|\{(g, x) \in G \times X \mid g \cdot x=x\}| \\ + & =\sum_{x \in X}\left|\mathrm{Stab}_G(x)\right| \\ + & =\sum_{x \in X}\frac{|G|}{\left|G x\right|} \quad \text{by Orbit-Stabilizer Theorem} \\ + & =|G| \sum_{G y \in G \backslash X}\; \sum_{x \in Gy}\frac{1}{\left|G x\right|} \\ + & =|G| \sum_{G y \in G \backslash X} \left|Gy\right|\frac{1}{\left|G y\right|} \\ + & =|G| \sum_{G y \in G \backslash X} 1 \\ + & =|G| \cdot|G \backslash X|. \end{align*} \end{prf} @@ -636,27 +662,27 @@ \subsection{Conjugacy Action} \begin{definition}{Conjugacy Action and Inner Automorphism Group}{conjugacy_action} Let $G$ be a group. The \textbf{conjugacy action} of $G$ on itself is defined as a group homomorphism \begin{align*} - \gamma:G &\longrightarrow \mathrm{Aut}_{\mathsf{Grp}}(G)\\ - g &\longmapsto (\gamma_g: x\longmapsto gxg^{-1}) + \gamma:G & \longrightarrow \mathrm{Aut}_{\mathsf{Grp}}(G) \\ + g & \longmapsto (\gamma_g: x\longmapsto gxg^{-1}) \end{align*} The \textbf{inner automorphism group} of $G$ is defined as the image of $\gamma$ $$ - \mathrm{Inn}(G)=\operatorname{im}\gamma=\{ \gamma_g\mid g\in G\}. - $$ + \mathrm{Inn}(G)=\operatorname{im}\gamma=\{ \gamma_g\mid g\in G\}. + $$ And we have inclusion relation $\mathrm{Inn}(G)\hookrightarrow\mathrm{Aut}_{\mathsf{Grp}}(G)\hookrightarrow\mathrm{Aut}_{\mathsf{Set}}(G)$. \end{definition} \begin{definition}{Conjugate Subgroups}{} -From \Cref{ex:acting_on_power_set}, we see conjugacy action on $G$ induces an action on its power set $2^G$: \begin{align*} - G\times 2^G &\longrightarrow 2^G\\ - (g,E)&\longmapsto gEg^{-1} -\end{align*} -If $H$ is a subgroup of $G$, then $gHg^{-1}$ is also a subgroup of $G$. We say $H$ and $gHg^{-1}$ are \textbf{conjugate subgroups} of $G$. + From \Cref{ex:acting_on_power_set}, we see conjugacy action on $G$ induces an action on its power set $2^G$: \begin{align*} + G\times 2^G & \longrightarrow 2^G \\ + (g,E) & \longmapsto gEg^{-1} + \end{align*} + If $H$ is a subgroup of $G$, then $gHg^{-1}$ is also a subgroup of $G$. We say $H$ and $gHg^{-1}$ are \textbf{conjugate subgroups} of $G$. \end{definition} -\begin{proposition}{Equivalent -Characterization of Inner Automorphisms}{} - Let $G$ be a group and $\varphi \in \mathrm{Aut}(G)$. Then $\varphi \in \mathrm{Inn}(G)$ if and only if $\varphi$ satisfies the property: +\begin{proposition}{Equivalent + Characterization of Inner Automorphisms}{} + Let $G$ be a group and $\varphi \in \mathrm{Aut}(G)$. Then $\varphi \in \mathrm{Inn}(G)$ if and only if $\varphi$ satisfies the property: \[ \text{$G$ is embedded in a group $H$} \implies \text{$\varphi$ extends to an automorphism of $H$}. \] @@ -665,7 +691,7 @@ \subsection{Conjugacy Action} \[ \begin{tikzcd}[ampersand replacement=\&] G \arrow[r, hook, "\iota"] \arrow[d, "\varphi"'] \& H \arrow[d, "\psi"] \\ - G \arrow[r, hook, "\iota"'] \& H + G \arrow[r, hook, "\iota"'] \& H \end{tikzcd} \] \end{proposition} @@ -674,54 +700,54 @@ \subsection{Conjugacy Action} \begin{definition}{Outer Automorphism Group}{} - Let $G$ be a group. Then we have $\mathrm{Inn}(G) \lhd \mathrm{Aut}_{\mathsf{Grp}}(G)$. And the \textbf{outer automorphism group} of $G$ is defined as + Let $G$ be a group. Then we have $\mathrm{Inn}(G) \lhd \mathrm{Aut}_{\mathsf{Grp}}(G)$. And the \textbf{outer automorphism group} of $G$ is defined as + $$ + \mathrm{Out}(G)=\mathrm{coker}\,\gamma=\mathrm{Aut}_{\mathsf{Grp}}(G)/\mathrm{Inn}(G). $$ - \mathrm{Out}(G)=\mathrm{coker}\,\gamma=\mathrm{Aut}_{\mathsf{Grp}}(G)/\mathrm{Inn}(G). - $$ \end{definition} \begin{definition}{Characteristic Subgroup}{} - Let $G$ be a group. A subgroup $H\le G$ is called a \textbf{characteristic subgroup} if + Let $G$ be a group. A subgroup $H\le G$ is called a \textbf{characteristic subgroup} if \[ \forall \varphi \in\mathrm{Aut}_{\mathsf{Grp}}(G),\; \varphi(H)\subseteq H. \] - It would be equivalent to require the stronger condition that $\forall \varphi \in\mathrm{Aut}_{\mathsf{Grp}}(G)$, $\varphi(H)= H$, because + It would be equivalent to require the stronger condition that $\forall \varphi \in\mathrm{Aut}_{\mathsf{Grp}}(G)$, $\varphi(H)= H$, because \[ \varphi(H)\subseteq H\implies \varphi^{-1}(H)\subseteq H\implies H\subseteq \varphi(H). \] \end{definition} \begin{definition}{Fully Characteristic Subgroup}{} - Let $G$ be a group. A subgroup $H\le G$ is called a \textbf{fully characteristic subgroup} if + Let $G$ be a group. A subgroup $H\le G$ is called a \textbf{fully characteristic subgroup} if \[ \forall \varphi \in\mathrm{End}_{\mathsf{Grp}}(G),\; \varphi(H)\subseteq H. \] \end{definition} \begin{definition}{Word Map}{} - Suppose $G$ is a group and + Suppose $G$ is a group and $$ - x=x_{i_1}^{\alpha_{1}}\cdots x_{i_m}^{\alpha_{m}}\in F\langle x_1,\cdots,x_n\rangle + x=x_{i_1}^{\alpha_{1}}\cdots x_{i_m}^{\alpha_{m}}\in F\langle x_1,\cdots,x_n\rangle $$ is a reduced word in a free group of rank $n$, where $\alpha_k\in\mathbb{Z}-\{0\}$ for $k=1,2,\cdots,m$. The \textbf{word map} induced by $x$ is defined as a map \begin{align*} - w_x:G^m &\longrightarrow G\\ - (g_1,\cdots,g_m) &\longmapsto g_{i_1}^{\alpha_1}\cdots g_{i_m}^{\alpha_m}. + w_x:G^m & \longrightarrow G \\ + (g_1,\cdots,g_m) & \longmapsto g_{i_1}^{\alpha_1}\cdots g_{i_m}^{\alpha_m}. \end{align*} \end{definition} \begin{definition}{Verbal Subgroup}{} Let $G$ be a group and $\mathcal{W}$ be a collection of word maps. A subgroup $H\le G$ is called a \textbf{verbal subgroup} if $H$ is the subgroup generated by $$ - \left\{ w(g_1,\cdots,g_n)\mid w\in\mathcal{W},\; g_i\in G \right\}. + \left\{ w(g_1,\cdots,g_n)\mid w\in\mathcal{W},\; g_i\in G \right\}. $$ \end{definition} \begin{definition}{Commutator}{} Let $G$ be a group. The word map induced by $xyx^{-1}y^{-1}$ is a binary operation defined on $G$, denoted by \begin{align*} - [\cdot,\cdot]:G\times G &\longrightarrow G\\ - (x,y) &\longmapsto [x,y]=xyx^{-1}y^{-1} + [\cdot,\cdot]:G\times G & \longrightarrow G \\ + (x,y) & \longmapsto [x,y]=xyx^{-1}y^{-1} \end{align*} $[x,y]$ is called the \textbf{commutator} of $x$ and $y$. \end{definition} @@ -746,7 +772,7 @@ \subsection{Conjugacy Action} \begin{definition}{Commutator Subgroup}{} Let $G$ be a group. The \textbf{commutator subgroup} or \textbf{derived subgroup} of $G$ is the subgroup generated by all the commutators, denoted by $$ - [G,G]=\langle \left\{[x,y]\mid x,y\in G \right\}\rangle. + [G,G]=\langle \left\{[x,y]\mid x,y\in G \right\}\rangle. $$ \end{definition} @@ -763,25 +789,25 @@ \subsection{Conjugacy Action} \begin{definition}{Abelianization}{} Let $G$ be a group. The \textbf{abelianization} of $G$ is defined as the quotient group $$ - G^{\mathrm{ab}}=G/[G,G]. + G^{\mathrm{ab}}=G/[G,G]. $$ \end{definition} \begin{proposition}{Universal Property of Abelianization}{} Let $G$ be a group and $A$ be an abelian group. Then any group homomorphism $f:G\to A$ factors through $G^{\mathrm{ab}}$ uniquely, that is, there exists a unique homomorphism $\bar{f}:G^{\mathrm{ab}}\to A$ such that the following diagram commutes $$ - \begin{tikzcd}[ampersand replacement=\&] - G \arrow[rr, "f"] \arrow[rd] \& \& A \\ - \& G^{\mathrm{ab}} \arrow[ru, "\bar{f}"'] \& - \end{tikzcd} + \begin{tikzcd}[ampersand replacement=\&] + G \arrow[rr, "f"] \arrow[rd] \& \& A \\ + \& G^{\mathrm{ab}} \arrow[ru, "\bar{f}"'] \& + \end{tikzcd} $$ \end{proposition} \begin{definition}{Normalizer}{} - Let $G$ be a group and $S$ be a subset of $G$. The \textbf{normalizer} of $S$ in $G$ is defined as + Let $G$ be a group and $S$ be a subset of $G$. The \textbf{normalizer} of $S$ in $G$ is defined as + $$ + \mathrm{N}_G(S)=\left\{ g\in G\mid gSg^{-1}=S \right\}. $$ - \mathrm{N}_G(S)=\left\{ g\in G\mid gSg^{-1}=S \right\}. - $$ Let $G$ acts on $2^G$ by conjugation (c.f. \Cref{th:conjugacy_action}). Then $\mathrm{N}_G(S)=\mathrm{Stab}_G(S)\le G$. \end{definition} @@ -794,13 +820,13 @@ \subsection{Conjugacy Action} \end{prf} \begin{definition}{Centralizer}{} - Let $G$ be a group and $S$ be a subset of $G$. The \textbf{centralizer} of $S$ in $G$ is defined as + Let $G$ be a group and $S$ be a subset of $G$. The \textbf{centralizer} of $S$ in $G$ is defined as \[ \mathrm{C}_G(S)=\left\{g\in G\mid \forall s\in S,\;gs=sg\right\}. \] - The centralizer of $\{x\}$ is the stabilizer subgroup of $x$ under conjugacy action, denoted by + The centralizer of $\{x\}$ is the stabilizer subgroup of $x$ under conjugacy action, denoted by \[ - \mathrm{C}_G(x)=\{ g\in G\mid gx=xg \}=\{ g\in G\mid gxg^{-1}=x \}= \mathrm{Stab}_G(x)=\mathrm{N}_G(x). + \mathrm{C}_G(x)=\{ g\in G\mid gx=xg \}=\{ g\in G\mid gxg^{-1}=x \}= \mathrm{Stab}_G(x)=\mathrm{N}_G(x). \] \end{definition} @@ -819,28 +845,28 @@ \subsection{Conjugacy Action} \begin{proposition}{Normalizer $\mathrm{N}_G(S)$ Acts on $S$ by Conjugation}{normalizer_conjugation_action} Let $G$ be a group and $S\subseteq G$. Then $\mathrm{N}_G(S)$ acts on $S$ by conjugation, i.e. by the group homomorphism \begin{align*} - \Psi_S: \mathrm{N}_G(S) &\longrightarrow \mathrm{Aut}_{\mathsf{Set}}(S)\\ - g &\longmapsto \gamma_g|_S + \Psi_S: \mathrm{N}_G(S) & \longrightarrow \mathrm{Aut}_{\mathsf{Set}}(S) \\ + g & \longmapsto \gamma_g|_S \end{align*} where $\gamma_g|_S(s)=gsg^{-1}$ for all $s\in S$. Moreover, we have $\ker\Psi_S=\mathrm{C}_G(S)\lhd \mathrm{N}_G(S)$. \end{proposition} \proof{ - $\Psi_S$ is obtained from restriction $\Psi_S:\mathrm{N}_G(S)\hookrightarrow G\xrightarrow{\gamma}\mathrm{Aut}_{\mathsf{Set}}(G)$. Since for any $g\in \mathrm{N}_G(S)$, $\gamma_g|_S(S)=\left\{gsg^{-1}\in G\mid s \in S\right\}\subseteq S$, we see $\Psi_S(g)=\gamma_g|_S\in \mathrm{Aut}_{\mathsf{Set}}(S)$. The kernel of $\Psi_S$ is - \begin{align*} - \ker\Psi_S &= \left\{ n\in \mathrm{N}_G(S)\mid \gamma_n|_S=\mathrm{id}_S \right\}= \left\{ n\in \mathrm{N}_G(S)\mid \forall s\in S, nsn^{-1}=s \right\}=\mathrm{N}_G(S)\cap \mathrm{C}_G(S) = \mathrm{C}_G(S). - \end{align*} +$\Psi_S$ is obtained from restriction $\Psi_S:\mathrm{N}_G(S)\hookrightarrow G\xrightarrow{\gamma}\mathrm{Aut}_{\mathsf{Set}}(G)$. Since for any $g\in \mathrm{N}_G(S)$, $\gamma_g|_S(S)=\left\{gsg^{-1}\in G\mid s \in S\right\}\subseteq S$, we see $\Psi_S(g)=\gamma_g|_S\in \mathrm{Aut}_{\mathsf{Set}}(S)$. The kernel of $\Psi_S$ is +\begin{align*} + \ker\Psi_S & = \left\{ n\in \mathrm{N}_G(S)\mid \gamma_n|_S=\mathrm{id}_S \right\}= \left\{ n\in \mathrm{N}_G(S)\mid \forall s\in S, nsn^{-1}=s \right\}=\mathrm{N}_G(S)\cap \mathrm{C}_G(S) = \mathrm{C}_G(S). +\end{align*} } \begin{theorem}{N/C Theorem}{N/C_theorem} - Let $G$ be a group and $H$ be a subgroup of $G$. By \Cref{th:normalizer_conjugation_action}, $\mathrm{N}_G(H)$ acts on $H$ by conjugation through the group homomorphism $\Psi_H:\mathrm{N}_G(H)\to \mathrm{Aut}_{\mathsf{Set}}(H)$. We assert that + Let $G$ be a group and $H$ be a subgroup of $G$. By \Cref{th:normalizer_conjugation_action}, $\mathrm{N}_G(H)$ acts on $H$ by conjugation through the group homomorphism $\Psi_H:\mathrm{N}_G(H)\to \mathrm{Aut}_{\mathsf{Set}}(H)$. We assert that \[ \mathrm{N}_G(H)/\mathrm{C}_G(H) \cong \mathrm{Im}\Psi_H\le \mathrm{Aut}_{\mathsf{Grp}}(H). \] Hence it is legal to define \begin{align*} - \Psi_H: \mathrm{N}_G(H) &\longrightarrow \mathrm{Aut}_{\mathsf{Grp}}(H)\\ - g &\longmapsto \gamma_g|_H + \Psi_H: \mathrm{N}_G(H) & \longrightarrow \mathrm{Aut}_{\mathsf{Grp}}(H) \\ + g & \longmapsto \gamma_g|_H \end{align*} \end{theorem} @@ -852,7 +878,7 @@ \subsection{Conjugacy Action} \end{corollary} \begin{prf} -Take $H=G$ in \Cref{th:N/C_theorem} we can get $G/Z_G\cong \mathrm{Inn}(G)$. + Take $H=G$ in \Cref{th:N/C_theorem} we can get $G/Z_G\cong \mathrm{Inn}(G)$. \end{prf} @@ -877,17 +903,17 @@ \subsection{Conjugacy Action} \begin{prf} \begin{enumerate}[(i)] \item According to \Cref{th:kernel_cokernel_of_conjugation} and \Cref{th:properties_of_stabilizer_subgroup} (ii), we have - \[ - x\in Z_G\iff x\in \ker \gamma\iff x\in - \bigcap\limits_{g\in G}\mathrm{C}_G(g). - \] + \[ + x\in Z_G\iff x\in \ker \gamma\iff x\in + \bigcap\limits_{g\in G}\mathrm{C}_G(g). + \] \end{enumerate} \end{prf} \begin{definition}{Conjugacy Class}{} - Let $G$ be a group. The orbit of $a$ under conjugacy action is called the \textbf{conjugacy class} of $a$, denoted by + Let $G$ be a group. The orbit of $a$ under conjugacy action is called the \textbf{conjugacy class} of $a$, denoted by \[ - \mathrm{Cl}(a)=\{ gag^{-1}\mid x\in G \}. + \mathrm{Cl}(a)=\{ gag^{-1}\mid x\in G \}. \] Two elements $a,b\in G$ are called \textbf{conjugate} if $\mathrm{Cl}(a)=\mathrm{Cl}(b)$. \end{definition} @@ -895,25 +921,25 @@ \subsection{Conjugacy Action} \begin{proposition}{Conjugacy Class of Element of Center is Singleton}{} Consider a group $G$ under conjugacy action. The $G$-invariant elements under conjugacy action are elements in the center of $G$: - \[ + \[ a\in Z_G\iff \mathrm{Cl}(a)=\{ a\}\iff \mathrm{C}_G(a)=\mathrm{Stab}_G(a)=G. \] \end{proposition} \begin{prf} According to \Cref{th:properties_of_stabilizer_subgroup} (i), $\mathrm{Cl}(a)=\{ a\}\iff \mathrm{Stab}_G(a)=G$. $$ - \begin{aligned} - a \in Z_G \iff&\forall g \in G,\;g a =a g \\ - \iff&\forall g \in G,\;g a g^{-1} =a \\ - \iff&\mathrm{C}_G(a) =\{a\} - \end{aligned} + \begin{aligned} + a \in Z_G \iff & \forall g \in G,\;g a =a g \\ + \iff & \forall g \in G,\;g a g^{-1} =a \\ + \iff & \mathrm{C}_G(a) =\{a\} + \end{aligned} $$ \end{prf} \begin{proposition}{Conjugacy Class Equation}{} Suppose $G$ is a finite group. If the distinct conjugacy classes of $G$ which are not singletons are $\mathrm{Cl}(x_1),\cdots,\mathrm{Cl}(x_m)$, then we have the \textbf{conjugacy class equation} \[ - |G|=|Z(G)|+\sum_{j=1}^m\left[G: Z_G\left(x_j\right)\right] + |G|=|Z(G)|+\sum_{j=1}^m\left[G: Z_G\left(x_j\right)\right] \] \end{proposition} \begin{prf} @@ -928,37 +954,37 @@ \subsection{Conjugacy Action} \end{proposition} \begin{prf} - For any $n \in \mathrm{N}_G(H)$ with image $\bar{n} \in \mathrm{N}_G(H)/H$, since $G$ acts transitively on $S$, we can define a map + For any $n \in \mathrm{N}_G(H)$ with image $\bar{n} \in \mathrm{N}_G(H)/H$, since $G$ acts transitively on $S$, we can define a map \begin{align*} - \phi(\bar{n}):S &\longrightarrow S\\ - g\cdot s &\longmapsto gn^{-1}\cdot s + \phi(\bar{n}):S & \longrightarrow S \\ + g\cdot s & \longmapsto gn^{-1}\cdot s \end{align*} and check $\phi(\bar{n})\in\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$ by \begin{itemize} - \item $\phi(\bar{n})\in\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$. $\phi(\bar{n})$ is well-defined: - \begin{align*} - &\bar{m}=\bar{n}\in\mathrm{N}_G(H)/H\\ - \implies &m^{-1}\in n^{-1}H\\ - \implies & \exists h\in H,m^{-1}=n^{-1}h\\ - \implies&\phi(\bar{m})(s)=gm^{-1}\cdot s=gn^{-1}h\cdot s=gn^{-1}\cdot s=\phi(\bar{n})(s). - \end{align*} + \item $\phi(\bar{n})\in\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$. $\phi(\bar{n})$ is well-defined: + \begin{align*} + & \bar{m}=\bar{n}\in\mathrm{N}_G(H)/H \\ + \implies & m^{-1}\in n^{-1}H \\ + \implies & \exists h\in H,m^{-1}=n^{-1}h \\ + \implies & \phi(\bar{m})(s)=gm^{-1}\cdot s=gn^{-1}h\cdot s=gn^{-1}\cdot s=\phi(\bar{n})(s). + \end{align*} \item $\phi(\bar{n})$ is a $G$-set morphism: $\phi(\bar{n})(g\cdot s)=gn^{-1}\cdot s=g\cdot\phi(\bar{n})(s)$ \item $\phi(\bar{n})$ is a bijection: $\phi(\bar{n}^{-1})$ is the inverse of $\phi(\bar{n})$. \end{itemize} - For any automorphism $\psi\in \mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$, by transitivity we have $\psi(s)=n^{-1}\cdot s$ for some $n \in G$. For $h \in H$, $hn^{-1}\cdot s=h\cdot\psi(s)=\psi(h\cdot s)=\psi(s)=n^{-1}\cdot s$, hence $n h n^{-1} \in H$ and $n \in \mathrm{N}_G(H)$. This gives a well-defined map + For any automorphism $\psi\in \mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$, by transitivity we have $\psi(s)=n^{-1}\cdot s$ for some $n \in G$. For $h \in H$, $hn^{-1}\cdot s=h\cdot\psi(s)=\psi(h\cdot s)=\psi(s)=n^{-1}\cdot s$, hence $n h n^{-1} \in H$ and $n \in \mathrm{N}_G(H)$. This gives a well-defined map \begin{align*} - \eta:\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S) &\longrightarrow \mathrm{N}_G(H)/H\\ - \psi &\longmapsto \bar{n} + \eta:\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S) & \longrightarrow \mathrm{N}_G(H)/H \\ + \psi & \longmapsto \bar{n} \end{align*} Clearly $\eta(\phi(\bar{n}))=\bar{n}$. Suppose $\psi\in\mathrm{Aut}_{G\text{-}\mathsf{Set}}(S)$ and $\psi(s)=n^{-1}\cdot s$. Then $\phi(\eta(\psi))(g\cdot s)=g n^{-1}\cdot s=g\psi(s)=\psi(g\cdot s)$, which imples $\phi(\eta(\psi))=\psi$. Therefore, $\phi:\mathrm{N}_G(H)/H\to\mathrm{Aut}_{G\text{-}\mathsf{Set}}$ is bijective. And it is easy to check that this $\phi$ is an isomorphism of groups, \[ - \phi(\bar{n}\bar{m})(g\cdot s)=\phi(\overline{nm})(g\cdot s)=g(nm)^{-1}\cdot s=gm^{-1}n^{-1}\cdot s=\phi(\bar{n})(gm^{-1}\cdot s)=\phi(\bar{n})\circ\phi(\bar{m})(g\cdot s). + \phi(\bar{n}\bar{m})(g\cdot s)=\phi(\overline{nm})(g\cdot s)=g(nm)^{-1}\cdot s=gm^{-1}n^{-1}\cdot s=\phi(\bar{n})(gm^{-1}\cdot s)=\phi(\bar{n})\circ\phi(\bar{m})(g\cdot s). \] \end{prf} -\begin{proposition}{}{} - A $G$-map $\alpha: G / H \longrightarrow G / K$ has the form $\alpha(g H)=g r K$, where the element $r \in G$ satisfies $r^{-1} H r\subseteq K$. +\begin{proposition}{$G$-maps between Left Coset Spaces}{} + Suppose $H,K$ are subgroups of $G$. Then we know $G$ left acts on $G/H$ and $G/K$. A $G$-map $\alpha: G / H \longrightarrow G / K$ has the form $\alpha(g H)=g r K$, where the element $r \in G$ satisfies $r^{-1} H r\subseteq K$. \end{proposition} \begin{prf} @@ -966,11 +992,11 @@ \subsection{Conjugacy Action} \[ \alpha(g H)=\alpha(g 1_G H)=g \alpha(1_G H). \] -Taking $\alpha(1_G H)=r K$, then we obtain $\alpha(g H)=g r K$. Since for all $h\in H$ we have -$$ -r K=\alpha(h1_G H)=h \alpha(1_G H)=hr K\implies r^{-1} h r \in K, -$$ -we show $r^{-1} H r\subseteq K$. + Taking $\alpha(1_G H)=r K$, then we obtain $\alpha(g H)=g r K$. Since for all $h\in H$ we have + $$ + r K=\alpha(h1_G H)=h \alpha(1_G H)=hr K\implies r^{-1} h r \in K, + $$ + we show $r^{-1} H r\subseteq K$. \end{prf} @@ -996,93 +1022,93 @@ \section{Symmetric Groups} \begin{theorem}{Pólya Enumeration Theorem (Unweighted)}{Polya_enumeration_unweighted} Let $X, Y$ be finite sets, where $X=\{1,2,\cdots,n\}$ is the set of points to be colored and $Y$ is the set of colors. Suppose a group $G$ acts on $X$ through $\sigma:G\to\mathrm{Aut}_{\mathsf{Set}}(X)$. Then it also acts on $Y^X$ by \Cref{ex:acting_on_functions}. Define that a coloring configuration of $(X,Y,\sigma)$ is an orbit in $Y^X/G$. Then the number of essentially distinct coloring configurations is -$$ -\left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}|Y|^{c(g)}, -$$ -where $c(g):=\left|\langle g\rangle \backslash X\right|=\left|\langle \sigma_g\rangle \backslash X\right|$ denotes the number of cycles in the cycle decomposition of $\sigma_g \in S_n$. + $$ + \left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}|Y|^{c(g)}, + $$ + where $c(g):=\left|\langle g\rangle \backslash X\right|=\left|\langle \sigma_g\rangle \backslash X\right|$ denotes the number of cycles in the cycle decomposition of $\sigma_g \in S_n$. \end{theorem} -\begin{prf} +\begin{prf} We apply Burnside's lemma \ref{th:Burnside's_lemma}, which states that -$$ -\left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}\left|\left(Y^X\right)^g\right| -$$ -where -\begin{align*} - \left(Y^X\right)^g&=\left\{ f \in Y^X \mid f(g \cdot x)=f (x)\text{ for all }x \in X\right\}\\ - &=\left\{ f \in Y^X \mid \langle g\rangle x = \langle g\rangle y\implies f(x)=f(y)\right\}. -\end{align*} -Define an equivalence relation $\sim_g$ on $X$ by $x \sim_g y\iff\langle g\rangle x = \langle g\rangle y$. Then we have $X/\sim_g\;=\langle g\rangle \backslash X$. By the universal property of the quotient set, for any map $f:X\to Y$ satisfying $x\sim y\implies f(x)=f(y)$, there exists a unique map $\overline{f}:\langle \sigma_g\rangle \backslash X\to Y$ such that the following diagram commutes -\[ -\begin{tikzcd}[ampersand replacement=\&] - X \arrow[rr, "f"] \arrow[rd,"\pi"'] \& \& Y \\ - \& \langle \sigma_g\rangle \backslash X \arrow[ru, "\overline{f}"',dashed] \& -\end{tikzcd} -\] -And we have a natural bijection between $\left(Y^X\right)^g$ and $Y^{\langle g\rangle \backslash X}$, which implies that $\left|\left(Y^X\right)^g\right|=|Y|^{c(g)}$. + $$ + \left|Y^X / G\right|=\frac{1}{|G|} \sum_{g \in G}\left|\left(Y^X\right)^g\right| + $$ + where + \begin{align*} + \left(Y^X\right)^g & =\left\{ f \in Y^X \mid f(g \cdot x)=f (x)\text{ for all }x \in X\right\} \\ + & =\left\{ f \in Y^X \mid \langle g\rangle x = \langle g\rangle y\implies f(x)=f(y)\right\}. + \end{align*} + Define an equivalence relation $\sim_g$ on $X$ by $x \sim_g y\iff\langle g\rangle x = \langle g\rangle y$. Then we have $X/\sim_g\;=\langle g\rangle \backslash X$. By the universal property of the quotient set, for any map $f:X\to Y$ satisfying $x\sim y\implies f(x)=f(y)$, there exists a unique map $\overline{f}:\langle \sigma_g\rangle \backslash X\to Y$ such that the following diagram commutes + \[ + \begin{tikzcd}[ampersand replacement=\&] + X \arrow[rr, "f"] \arrow[rd,"\pi"'] \& \& Y \\ + \& \langle \sigma_g\rangle \backslash X \arrow[ru, "\overline{f}"',dashed] \& + \end{tikzcd} + \] + And we have a natural bijection between $\left(Y^X\right)^g$ and $Y^{\langle g\rangle \backslash X}$, which implies that $\left|\left(Y^X\right)^g\right|=|Y|^{c(g)}$. \end{prf} \begin{definition}{Cycle Index Polynomial}{} Let $G$ be a subgroup of $S_n$. The \textbf{cycle index polynomial} of $G$ is defined as $$ - Z(t_1,t_2,\cdots,t_n;G)=\frac{1}{|G|}\sum_{g\in G}t_1^{c_1(g)}t_2^{c_2(g)}\cdots t_n^{c_n(g)}, + Z(t_1,t_2,\cdots,t_n;G)=\frac{1}{|G|}\sum_{g\in G}t_1^{c_1(g)}t_2^{c_2(g)}\cdots t_n^{c_n(g)}, $$ where $c_k(g)$ denotes the number of $k$-cycles in the cycle decomposition of $\sigma_g$. $|G|Z(t_1,t_2,\cdots,t_n;G)$ can be seen as a generating function, where the coefficient of $t_1^{c_1}t_2^{c_2}\cdots t_n^{c_n}$ represents the number of permutations in $G$ with excatly $c_k$ $k$-cycles for $k=1,2,\cdots,n$. \end{definition} \begin{theorem}{Pólya Enumeration Theorem (Weighted)}{} -Let $X, Y$ be finite sets, where $X=\{1,2,\cdots,n\}$ is the set of points to be colored and $Y$ is the set of colors. Suppose $w:Y\to\mathbb{Z}_{\ge 0}^m$ is a weight function which assigns a weight $w(y)=(w_1(y),w_2(y),\cdots,w_m(y))$ to each color $y\in Y$. Consider the genrating function -\[ -q(x_1,\cdots,x_m)=\sum_{y\in Y}x_1^{w_1(y)}x_2^{w_2(y)}\cdots x_m^{w_m(y)}, -\] -where the coefficient of the term $x_1^{a_1}x_2^{a_2}\cdots x_m^{a_m}$ is the number of colors with weight $(a_1,a_2,\cdots,a_m)$. For each coloring map $f:X\to Y$, define the its weight as $W(f)$, where -\begin{align*} - W:Y^X&\longrightarrow\mathbb{Z}_{\ge 0}^m\\ -f&\longmapsto\sum_{x\in X}w(f(x)) -\end{align*} -Let $G$ be a subgroup of $S_n$. Given any $g\in G$ and $f\in X^Y$, we can check the action of $g$ on $f$ does not change its weight -\begin{align*} - W(f\star g)=\sum_{x\in X}w(( f\star g)(x))=\sum_{x\in X}w(f(g\cdot x))=\sum_{x\in X}w(f(x))=W(f), -\end{align*} -which implies that for each $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$, the fiber $W^{-1}(\omega)$ is $G$-invariant. + Let $X, Y$ be finite sets, where $X=\{1,2,\cdots,n\}$ is the set of points to be colored and $Y$ is the set of colors. Suppose $w:Y\to\mathbb{Z}_{\ge 0}^m$ is a weight function which assigns a weight $w(y)=(w_1(y),w_2(y),\cdots,w_m(y))$ to each color $y\in Y$. Consider the genrating function + \[ + q(x_1,\cdots,x_m)=\sum_{y\in Y}x_1^{w_1(y)}x_2^{w_2(y)}\cdots x_m^{w_m(y)}, + \] + where the coefficient of the term $x_1^{a_1}x_2^{a_2}\cdots x_m^{a_m}$ is the number of colors with weight $(a_1,a_2,\cdots,a_m)$. For each coloring map $f:X\to Y$, define the its weight as $W(f)$, where + \begin{align*} + W:Y^X & \longrightarrow\mathbb{Z}_{\ge 0}^m \\ + f & \longmapsto\sum_{x\in X}w(f(x)) + \end{align*} + Let $G$ be a subgroup of $S_n$. Given any $g\in G$ and $f\in X^Y$, we can check the action of $g$ on $f$ does not change its weight + \begin{align*} + W(f\star g)=\sum_{x\in X}w(( f\star g)(x))=\sum_{x\in X}w(f(g\cdot x))=\sum_{x\in X}w(f(x))=W(f), + \end{align*} + which implies that for each $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$, the fiber $W^{-1}(\omega)$ is $G$-invariant. -Define a coloring configuration of $(X,Y,G)$ as an orbit in $Y^X/G$. The generating function for the number of essentially distinct coloring configurations with weight $\omega$ can be expressed as -\begin{align*} - \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ - &= Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). -\end{align*} + Define a coloring configuration of $(X,Y,G)$ as an orbit in $Y^X/G$. The generating function for the number of essentially distinct coloring configurations with weight $\omega$ can be expressed as + \begin{align*} + \mathrm{CGF}\left(x_1, \cdots,x_m\right) & =\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m} \\ + & = Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). + \end{align*} \end{theorem} -\begin{prf} +\begin{prf} For any $\omega=(\omega_1,\cdots,\omega_m)\in\mathbb{Z}_{\ge 0}^m$, by applying Burnside's lemma \ref{th:Burnside's_lemma} to the $G$-set $W^{-1}(\omega)$, we have \[ \left|G\backslash W^{-1}(\omega)\right| =\frac{1}{|G|}\sum_{g\in G}\left|W^{-1}(\omega)^g\right|. \] - Similar to \Cref{th:Polya_enumeration_unweighted}, we have a bijection between + Similar to \Cref{th:Polya_enumeration_unweighted}, we have a bijection between \[ W^{-1}(\omega)^g=\left\{f\in \left(Y^X\right)^g\mid W(f)=\omega\right\}\quad\text{and}\quad\left\{\overline{f}\in Y^{\langle g\rangle\backslash X}\mid W\left(\,\overline{f}\circ\pi\right)=\omega\right\}, \] Suppose $\langle g\rangle\backslash X=\{\langle g\rangle x_1,\cdots,\langle g\rangle x_r\}$. We can give a coloring configuration by $r$ consecutive steps. In the $i$-th step, we just need to choose a color in $y\in Y$ for the $i$-orbit $\langle g\rangle\backslash x_i$, which will contribute a term - $$ - x_1^{|\langle g\rangle x_i| w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i| w_m(y)}. - $$ + $$ + x_1^{|\langle g\rangle x_i| w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i| w_m(y)}. + $$ Thus we have \begin{align*} - \sum_{\omega\in\mathbb{Z}_{\ge 0}^m} \left |W^{-1}(\omega)^g\right |x_1^{\omega_1} \cdots x_m^{\omega_m} &= \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X }\left(\sum_{y\in Y}x_1^{|\langle g\rangle x_i|w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i|w_m(y)}\right) \\ - &= \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X } q\left (x_1^{|\langle g\rangle x_i|}, x_2^{|\langle g\rangle x_i|}, \cdots,x_m^{|\langle g\rangle x_i|}\right )\\ -&= q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)}. + \sum_{\omega\in\mathbb{Z}_{\ge 0}^m} \left |W^{-1}(\omega)^g\right |x_1^{\omega_1} \cdots x_m^{\omega_m} & = \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X }\left(\sum_{y\in Y}x_1^{|\langle g\rangle x_i|w_1(y)}x_2^{|\langle g\rangle x_i|w_2(y)}\cdots x_m^{|\langle g\rangle x_i|w_m(y)}\right) \\ + & = \prod_{\langle g\rangle x_i \in \langle g\rangle\backslash X } q\left (x_1^{|\langle g\rangle x_i|}, x_2^{|\langle g\rangle x_i|}, \cdots,x_m^{|\langle g\rangle x_i|}\right ) \\ + & = q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)}. \end{align*} - where + where + $$ + c_k(g)=\sum_{\langle g\rangle x_i \in \langle g\rangle\backslash X}\mathbf{1}_{|\langle g\rangle x_i |=k} $$ - c_k(g)=\sum_{\langle g\rangle x_i \in \langle g\rangle\backslash X}\mathbf{1}_{|\langle g\rangle x_i |=k} - $$ denotes the number of orbits with size $k$. With this equality, we can rewrite the generating function as \begin{align*} - \mathrm{CGF}\left(x_1, \cdots,x_m\right) &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ - &=\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\frac{1}{|G|}\sum_{g \in G}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ - &=\frac{1}{|G|}\sum_{g \in G}\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m}\\ - &=\frac{1}{|G|}\sum_{g \in G}q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)}\\ - &= Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). + \mathrm{CGF}\left(x_1, \cdots,x_m\right) & =\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|G\backslash W^{-1}(\omega)\right| x_1^{\omega_1} \cdots x_m^{\omega_m} \\ + & =\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\frac{1}{|G|}\sum_{g \in G}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m} \\ + & =\frac{1}{|G|}\sum_{g \in G}\sum_{\omega \in \mathbb{Z}_{\ge 0}^m}\left|W^{-1}(\omega)^g\right| x_1^{\omega_1} \cdots x_m^{\omega_m} \\ + & =\frac{1}{|G|}\sum_{g \in G}q(x_1, \cdots,x_m)^{c_1(g)} q \left (x_1^2, \cdots,x_m^2 \right )^{c_2(g)} \cdots q \left (x_1^n, \cdots,x_m^n\right )^{c_n(g)} \\ + & = Z\left(q\left(x_1, \cdots,x_m\right), q\left(x_1^2, \cdots,x_m^2\right), \cdots, q\left(x_1^n, \cdots,x_m^n\right);G\right). \end{align*} \end{prf} @@ -1100,28 +1126,28 @@ \section{Symmetric Groups} \end{itemize} Writing out \(Z(t_1,\cdots,t_6;D_{12})\): -\[ - Z(t_1,\cdots,t_6;D_{12})=\frac{1}{12}\left(t_1^6+3 t_1^2 t_2^2+4 t_2^3+2 t_3^2+2 t_6\right) -\] + \[ + Z(t_1,\cdots,t_6;D_{12})=\frac{1}{12}\left(t_1^6+3 t_1^2 t_2^2+4 t_2^3+2 t_3^2+2 t_6\right) + \] -Assigning a weight of (1,0) to H and a weight of (0,1) to Cl, the corresponding generating function is \(q(\text{H},\text{Cl})=\text{H}+\text{Cl}\). Finally, the generating function for the number of essentially distinct coloring configurations is: + Assigning a weight of (1,0) to H and a weight of (0,1) to Cl, the corresponding generating function is \(q(\text{H},\text{Cl})=\text{H}+\text{Cl}\). Finally, the generating function for the number of essentially distinct coloring configurations is: -\[ -\begin{aligned} -&\quad \mathrm{CGF}(\text{H},\text{Cl})\\ -& =\frac{1}{12}\left((\text{H}+\text{Cl})^6+3(\text{H}+\text{Cl})^2\left(\text{H}^2+\text{Cl}^2\right)^2+4\left(\text{H}^2+\text{Cl}^2\right)^3+2\left(\text{H}^3+\text{Cl}^3\right)^2+2\left(\text{H}^6+\text{Cl}^6\right)\right) \\ -& =\text{H}^6+\text{H}^5 \text{Cl}+3 \text{H}^4 \text{Cl}^2+3 \text{H}^3 \text{Cl}^3+3 \text{H}^2 \text{Cl}^4+\text{H} \text{Cl}^5+\text{Cl}^6 -\end{aligned} -\] + \[ + \begin{aligned} + & \quad \mathrm{CGF}(\text{H},\text{Cl}) \\ + & =\frac{1}{12}\left((\text{H}+\text{Cl})^6+3(\text{H}+\text{Cl})^2\left(\text{H}^2+\text{Cl}^2\right)^2+4\left(\text{H}^2+\text{Cl}^2\right)^3+2\left(\text{H}^3+\text{Cl}^3\right)^2+2\left(\text{H}^6+\text{Cl}^6\right)\right) \\ + & =\text{H}^6+\text{H}^5 \text{Cl}+3 \text{H}^4 \text{Cl}^2+3 \text{H}^3 \text{Cl}^3+3 \text{H}^2 \text{Cl}^4+\text{H} \text{Cl}^5+\text{Cl}^6 + \end{aligned} + \] -The coefficients give the number of isomers for various chlorobenzene compounds. For instance, looking at the term \(3 \text{H}^4 \text{Cl}^2\), with a weight of $(4,2)$, it can only be achieved using 4 H atoms and 2 Cl atoms. The coefficient 3 indicates that there are 3 isomers for dichlorobenzene. The 3 isomers are 1,2-dichlorobenzene, 1,3-dichlorobenzene, and 1,4-dichlorobenzene, plotted as follows: + The coefficients give the number of isomers for various chlorobenzene compounds. For instance, looking at the term \(3 \text{H}^4 \text{Cl}^2\), with a weight of $(4,2)$, it can only be achieved using 4 H atoms and 2 Cl atoms. The coefficient 3 indicates that there are 3 isomers for dichlorobenzene. The 3 isomers are 1,2-dichlorobenzene, 1,3-dichlorobenzene, and 1,4-dichlorobenzene, plotted as follows: -\begin{center} - \setchemfig{atom sep=1.5em} - \chemfig[scale=0.1]{*6(=-(-Cl)=(-Cl)-=-)} \qquad % 1,2-dichlorobenzene - \chemfig[scale=0.1]{*6(=-=(-Cl)-=(-Cl)-=)} \qquad % 1,3-dichlorobenzene - \chemfig[scale=0.1]{*6(=-(-Cl)=-=(-Cl)-)} % 1,4-dichlorobenzene -\end{center} + \begin{center} + \setchemfig{atom sep=1.5em} + \chemfig[scale=0.1]{*6(=-(-Cl)=(-Cl)-=-)} \qquad % 1,2-dichlorobenzene + \chemfig[scale=0.1]{*6(=-=(-Cl)-=(-Cl)-=)} \qquad % 1,3-dichlorobenzene + \chemfig[scale=0.1]{*6(=-(-Cl)=-=(-Cl)-)} % 1,4-dichlorobenzene + \end{center} \end{example} @@ -1137,25 +1163,25 @@ \section{Abelian Group} \begin{example}{Forgetful Functor $U_{\mathsf{Grp}}: \mathsf{Ab}\to \mathsf{Grp}$}{} The forgetful functor $U_{\mathsf{Grp}}: \mathsf{Ab}\to \mathsf{Grp}$ is a functor that sends an abelian group to its underlying group. It is a fully faithful functor. \end{example} - + \begin{example}{Forgetful Functor $U: \mathsf{Ab}\to \mathsf{Set}$}{} - The forgetful functor $U: \mathsf{Ab}\to \mathsf{Set}$ forgets the group structure and sends an abelian group to its underlying set. + The forgetful functor $U: \mathsf{Ab}\to \mathsf{Set}$ forgets the group structure and sends an abelian group to its underlying set. \begin{enumerate}[(i)] \item $U$ is representable by $\left(\mathbb{Z}, 1_\mathbb{Z}\right)$. \item $U$ is full but not faithful. \end{enumerate} \end{example} - + \begin{definition}{Free Abelian Group}{} A \textbf{free abelian group} generated by a set $X$ is an abelian group denoted by $\mathbb{Z}^{\oplus X}$ which satisfies the following universal property: for any group $G$ and any funtion $f:X\to G$, there exists a unique group homomorphism $\varphi:\mathbb{Z}^{\oplus X}\to G$ such that the following diagram commutes \[ - \begin{tikzcd}[ampersand replacement=\&] - \mathbb{Z}^{\oplus X} \arrow[r, "\exists!\varphi", dashed] \& G \\[+10pt] -X \arrow[u, "\iota"] \arrow[ru, "f"'] \& - \end{tikzcd} - \] + \begin{tikzcd}[ampersand replacement=\&] + \mathbb{Z}^{\oplus X} \arrow[r, "\exists!\varphi", dashed] \& G \\[+10pt] + X \arrow[u, "\iota"] \arrow[ru, "f"'] \& + \end{tikzcd} + \] where $\iota:X\to \mathbb{Z}^{\oplus X}$ is the canonical injection. \end{definition}