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--- problem:aut-square [2013/08/09 06:35]
joshuawiscons created
+++ problem:aut-square [2013/08/22 16:30] (current)
bmwoodruff
@@ Line 1: / Line 1: @@
 ====== Automorphism group of a square ======
-**Problem.** Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$, and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\]
-Write down all elements of $\textrm{Aut}(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers.
+==== Problem ====
-  - There some $a\in \textrm{Aut}(\mathcal{G})$ such that $a^2$ is the identity.
+Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$ and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\]
-  - There some $a\in \textrm{Aut}(\mathcal{G})$ such that $a^3$ is the identity.
+Write down all elements of $\aut(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers. For $a,b\in \aut(\mathcal{G})$, $ab$ denotes the (function) composition of $a$ with $b$, and $a^n$ denotes the composition of $a$ with itself $n$-times.
-  - There some $a\in \textrm{Aut}(\mathcal{G})$ such that every other $b\in  \textrm{Aut}(\mathcal{G})$ is of the form $b=a^n$ for some $n$.
+  - There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.
-  - For every $a,b\in \textrm{Aut}(\mathcal{G})$, $ab = ba$.
+  - There is an $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity but $a$ is not the identity.
+  - There is an $a\in \aut(\mathcal{G})$ such that every other $b\in  \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$.
+  - For every $a,b\in \aut(\mathcal{G})$, we have $ab = ba$.
 {{ wiki:graphpictures:labeledsquare.png?75 }}
@@ Line 11: / Line 13: @@
 ----
 ==== Remarks ====
-  * $\textrm{Aut}(\mathcal{G})$ denotes the [[definition:automorphism group]] of $\mathcal{G}$. The definition of an [[definition:automorphism group]] also explains the notation $a^n$ and $ab$.
+  * $\aut(\mathcal{G})$ denotes the [[definition:automorphism of a graph|automorphism group]] of $\mathcal{G}$.
   * $S_4$ denotes the [[definition:symmetric group]] on $\{1,2,3,4\}$.
 ----
 ==== $\LaTeX$ version ====
-<code>
+<file tex problem.aut-square.tex>
-%%%%%%%%%%
+%%%%%
 % DEPENDENCIES
-% RequiredPackages: \usepackage{tikz}
+% RequiredPackages \usepackage{tikz}
 % RequiredMacros: \DeclareMathOperator{\aut}{Aut}
-%%%%%%%%%%
+%%%%%
 \begin{problem}
-Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$, and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\]
+Consider the graph $\mathcal{G} = (V,E)$ drawn below. The vertex set is $V$ and the (symmetric) relation giving adjacency is $E$. Specifically, $V = \{1,2,3,4\}$ and \[E = \{(1,2),(2,1),(2,3),(3,2),(3,4),(4,3),(1,4),(4,1)\}.\]
-Write down all elements of $\aut(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers.
+Write down all elements of $\aut(\mathcal{G})$; view these as the elements of $S_4$ that preserve $E$. Also, determine if the following statements are true or false; explain your answers. For $a,b\in \aut(\mathcal{G})$, $ab$ denotes the (function) composition of $a$ with $b$, and $a^n$ denotes the composition of $a$ with itself $n$-times.
 \begin{enumerate}
-\item There some $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity.
+\item There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.
-\item There some $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity.
+\item There is an $a\in \aut(\mathcal{G})$ such that $a^3$ is the identity but $a$ is not the identity.
-\item There some $a\in \aut(\mathcal{G})$ such that every other $b\in  \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$.
+\item There is an $a\in \aut(\mathcal{G})$ such that every other $b\in  \aut(\mathcal{G})$ is of the form $b=a^n$ for some $n$.
 \item For every $a,b\in \aut(\mathcal{G})$, $ab = ba$.
 \end{enumerate}
@@ Line 37: / Line 39: @@
 \end{center}
 \end{problem}
-</code>
+</file>
 ----
 ==== External links ====
   * [[wp>Dihedral group]]
+{{tag>problem needsreview rjosh rben ben}}

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