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Exercise (Conjugacy Classes Of The Automorphisms Of The Square)

Let $G = D_8$, the automorphisms of the square. Compute the conjugacy classes of $G$.

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The conjugacy classes are $$ \begin{align} \text{cl}(R_0)&=\{R_0\}\\ \text{cl}(R_{180})&=\{R_{180}\}\\ \text{cl}(R_{90})&=\{R_{90},R_{270}\}\\ \text{cl}(R_{270})&=\{R_{90},R_{270}\}\\ \text{cl}(H)&=\{H,V\}\\ \text{cl}(V)&=\{H,V\}\\ \text{cl}(D)&=\{D,D'\}\\ \text{cl}(D-)&=\{D,D'\}. \end{align} $$ Notice how this partitions the group. The next problem has you show that conjugacy is an equivalence relation.

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Page last modified on December 10, 2013, at 05:12 PM

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HomePage Current Problems All Problems Pictures ProblemsByWeek List of Problems List of Definitions List of Theorems List of Conjectures All Recent changes (use this to find a page that may have been lost)	Exercise Conjugacy Classes Of The Automorphisms Of The Square Please Login to access more options. Exercise (Conjugacy Classes Of The Automorphisms Of The Square) Let $G = D_8$, the automorphisms of the square. Compute the conjugacy classes of $G$. Click to see a solution. The conjugacy classes are $$ \begin{align} \text{cl}(R_0)&=\{R_0\}\\ \text{cl}(R_{180})&=\{R_{180}\}\\ \text{cl}(R_{90})&=\{R_{90},R_{270}\}\\ \text{cl}(R_{270})&=\{R_{90},R_{270}\}\\ \text{cl}(H)&=\{H,V\}\\ \text{cl}(V)&=\{H,V\}\\ \text{cl}(D)&=\{D,D'\}\\ \text{cl}(D-)&=\{D,D'\}. \end{align} $$ Notice how this partitions the group. The next problem has you show that conjugacy is an equivalence relation. Edit Page History Source Attach File Backlinks List Group Page last modified on December 10, 2013, at 05:12 PM
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