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Typo fixes

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Mark 2024-01-08 10:56:22 -08:00
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12
Advanced/Symmetric Group/parts/1 cycle.tex
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@ -57,14 +57,14 @@ In other words, show that there is always an integer $n$ so that $f^n(x) = x$.
\vfill \vfill
\definition{Composition} \definition{Composition}<compdef>
The \textit{composition} of two permutations $f$ and $g$ is the permutation $h(x) = f(g(x))$. \par The \textit{composition} of two permutations $f$ and $g$ is the permutation $h(x) = f(g(x))$. \par
The usual notation for this is $f \circ g$, but we'll simply write $fg$. The usual notation for this is $f \circ g$, but we'll simply write $fg$.
\problem{} \problem{}
What is $[1324][4321]$? \par What is $[1324][4321]$? \par
How about $[321][213][231]$? \par How about $[321][213][231]$? \par
\hint{composition is left-associative, so we evaluate $abc$ as $(ab)c$} \hint{is composition is left or right-associative? See \ref{compdef}}
\vfill \vfill
@ -440,7 +440,7 @@ How about $(123)$? And $(4231)$? \par
\problem{} \problem{}
Say $\sigma$ is a permutation composed of cycles $\sigma_1\sigma_2...\sigma_k$. \par Say $\sigma$ is a permutation composed of disjoint cycles $\sigma_1\sigma_2...\sigma_k$. \par
Say we know the order of all $\sigma_i$. What is the order of $\sigma$? Say we know the order of all $\sigma_i$. What is the order of $\sigma$?
\begin{solution} \begin{solution}
@ -479,7 +479,8 @@ Show that any permutation is a product of transpositions.
Show that any permutation is a product of transpositions of the form $(1, k)$. \par Show that any permutation is a product of transpositions of the form $(1, k)$. \par
\begin{solution} \begin{solution}
Use \ref{simpletrans} and rewrite each $(a, b)$ as $(1, a)(1, b)(1, a)$. Use \ref{simpletrans} and rewrite each $(a, b)$ as $(1, a)(1, b)(1, a)$. \par
Showing that $(a, b) = (1, a)(1, b)(1, a)$ is fairly easy.
\end{solution} \end{solution}
\vfill \vfill
@ -491,7 +492,8 @@ Show that any permutation is a product of transpositions of the form $(1, k)$. \
Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$. Show that any transposition $(a, b)$ is equal to the product $(a, a+1)(a+1, b)(a, a+1)$.
\begin{solution} \begin{solution}
TODO This is the same as the $(1, a)(1, b)(1, a)$ case above, but we use $a + 1$
as a \say{working slot} instead of $1$.
\end{solution} \end{solution}
\vfill \vfill

2
Advanced/Symmetric Group/parts/3 subgroup.tex
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@ -98,7 +98,7 @@ Show that $\mathbb{Z}_7^\times$ and $\mathbb{Z}_9^\times$ are isomorphic.
\problem{} \problem{}
Show that $\mathbb{Z}_{10}^\times$ and $\mathbb{Z}_4^\times$, and $\mathbb{Z}_3$ are isomorphic. Show that $\mathbb{Z}_{10}^\times$ and $\mathbb{Z}_5^\times$, and $\mathbb{Z}_4$ are isomorphic.
\hint{ \hint{
Build a bijection with the above properties. \\ Build a bijection with the above properties. \\
Remember that a group is fully defined by its multiplication table. Remember that a group is fully defined by its multiplication table.