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problem:aut-square [2013/08/19 09:28]
joshuawiscons 		problem:aut-square [2013/08/22 16:30] (current)
bmwoodruff
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 ==== Problem ==== ==== 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.
   - There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.   - There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.
   - 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 $a^3$ is the identity but $a$ is not the identity.
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 ---- ----
 ==== Remarks ====   ==== Remarks ====  
-  * $\aut(\mathcal{G})$ 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 $\{1,2,3,4\}$.   * $S_4$ denotes the [[definition:symmetric group]] on $\{1,2,3,4\}$.
  
 ---- ----
 ==== $\LaTeX$ version ==== ==== $\LaTeX$ version ====
-<file tex aut-square.tex> +<file tex problem.aut-square.tex> 
-%%%%%%%%%%+%%%%%
 % DEPENDENCIES % DEPENDENCIES
-% RequiredPackages: \usepackage{tikz} +% RequiredPackages \usepackage{tikz} 
-% RequiredMacros: \DeclareMathOperator{\aut}{Aut}  +% RequiredMacros: \DeclareMathOperator{\aut}{Aut}   
-%%%%%%%%%%+%%%%%
 \begin{problem} \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} \begin{enumerate}
 \item There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity. \item There is an $a\in \aut(\mathcal{G})$ such that $a^2$ is the identity but $a$ is not the identity.
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   * [[wp>Dihedral group]]   * [[wp>Dihedral group]]
  
-{{tag>problem needsreview rjosh}}+{{tag>problem needsreview rjosh rben ben}}
problem/aut-square.1376918939.txt.gz · Last modified: 2013/08/19 09:28 by joshuawiscons

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