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problem:diagramatic_representation_of_sn

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problem:diagramatic_representation_of_sn [2013/08/21 07:18]
joshuawiscons created 		problem:diagramatic_representation_of_sn [2013/08/22 16:35] (current)
bmwoodruff
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-====== Diagramatic Representation Of Sn ======+====== Diagramatic Representation Of $S_n$ ======
 ==== Problem ==== ==== Problem ====
-The goal of this problem is to find a way to represent the symmetric group $S_n$ with diagrams. We will focus on $S_4$. +The goal of this problem is to find a way to represent the [[definition:symmetric group]] $S_n$ with diagrams. We will focus on $S_4$. 
-  - Which element of $S_4$ does the following diagram seem to represent? {{ problem:symfourdiagramacdb.png }}+  - Which element of $S_4$ does the following diagram seem to represent? {{ problem:symfourdiagramacdb.png?150 }}
   - What is the diagram for the inverse of the previous element.   - What is the diagram for the inverse of the previous element.
   - Formulate a diagrammatic rule for finding the diagram of the inverse of an element of $S_4$.   - Formulate a diagrammatic rule for finding the diagram of the inverse of an element of $S_4$.
   - What is the diagram for the identity.   - What is the diagram for the identity.
-  - Consider $\sigma,\tau\in S_4$ whose diagrams given below. Determine the diagrams for $\sigma\tau$ and $\sigma\tau$. {{ problem:symfourdiagramacdbbadc.png }}+  - Consider $\sigma,\tau\in S_4$ whose diagrams are given below. Determine the diagrams for $\sigma\tau$ and $\sigma\tau$. {{ problem:symfourdiagramacdbbadc.png?400 }}
   - Formulate a diagrammatic rule for finding the diagram of the composition of two elements of $S_4$.   - Formulate a diagrammatic rule for finding the diagram of the composition of two elements of $S_4$.
   - Find a $\sigma \in S_4$ such that $\sigma$ is not the identity but $\sigma^2$ is the identity. Find $5$ more (different) elements in $S_4$ with the same property.   - Find a $\sigma \in S_4$ such that $\sigma$ is not the identity but $\sigma^2$ is the identity. Find $5$ more (different) elements in $S_4$ with the same property.
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 ---- ----
 ==== Remarks ====   ==== Remarks ====  
-  * Make remarks with a list.+  * Answers to the problem, of course, depend on whether a diagram is read top-to-bottom or bottom-to-top and whether function composition is performed left-to-right or right-to-left.
  
 ---- ----
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 \item Formulate a diagrammatic rule for finding the diagram of the inverse of an element of $S_4$. \item Formulate a diagrammatic rule for finding the diagram of the inverse of an element of $S_4$.
 \item What is the diagram for the identity. \item What is the diagram for the identity.
-\item Consider $\sigma,\tau\in S_4$ whose diagrams given below. Determine the diagrams for $\sigma\tau$ and $\sigma\tau$.+\item Consider $\sigma,\tau\in S_4$ whose diagrams are given below. Determine the diagrams for $\sigma\tau$ and $\sigma\tau$.
 \[\sigma = \begin{tikzpicture}[baseline= (current bounding box.west),vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2] \[\sigma = \begin{tikzpicture}[baseline= (current bounding box.west),vertex/.style={circle,draw=black,fill=blue!15,thick,inner sep=1.5pt}, line width = 2]
 \foreach \i in {1,2,3,4} { \foreach \i in {1,2,3,4} {
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 ---- ----
 ==== External links ==== ==== External links ====
-  * None.+  * [[wp>Symmetric group]]
  
-{{tag>problem}}+{{tag>problem needsreview rjosh rben ben}}
problem/diagramatic_representation_of_sn.1377083907.txt.gz · Last modified: 2013/08/21 07:18 by joshuawiscons

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