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{\bf Math 113 Homework 6, due 3/7/2019}
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{\bf 1.} This is an important exercise, defining the group $D_n$. 

Let $C_n\subset S_n$ be the subset consisting of the following permutations: $$\{\rho_k = (1,2,3,\dots, n)^k, k=0,\dots, n-1\}.$$ Here remember that $(1,2,3,\dots, n)$ is the cyclic notation for the permutation $$\begin{pmatrix} 1,&2,&3,&\ldots,& n-2,& n-1,& n\\2,&3,&4,&\ldots,& n-1,&n,&1\end{pmatrix}.$$

  (a) Show that $C_n$ is a subgroup (hint: it's cyclic!)

  (b) Show that, if each $1,\dots, n$ is relabeled by its corresponding class in $\Z_n$ (remember that $[n]=[0]$), then $C_n$ consists of all functions of the type $\rho_{[k]}:\Z_n\to \Z_n$ for $[k]\in \Z_n,$ with $$\rho_{[k]}([x]) = [k]+[x]$$ defined to be the function that adds $[k]$ to any remainder modulo $n$. 

  (c) Let $\lambda\in S_n$ be the permutation given by $\lambda(k) = n+1-k$, so
  $$\lambda = \begin{pmatrix} 1,&2,&3,&\ldots,& n-2,& n-1,& n\\n,&n-1,&n-2,&\ldots,& 3,&2,&1\end{pmatrix}$$ in permutation notation.

    Show that $\lambda^2 = e$.

    (d) Define $\lambda_k = \rho_{k-1}\circ \lambda$. Show that if the indices $1,\dots, n$ are viewed as remainders modulo $n$ then $\lambda_{[k]}([x]) = [k]-[x].$

    (e) Give formulas for the four possible compositions, $\rho_{[k]}\circ \rho_{[k']}$, $\rho_{[k]}\circ \lambda_{[k']},$ $\lambda_{[k]}\circ \rho_{[k']}$ and $\lambda_{[k]}\circ \lambda_{[k']}$ (it is easier to do this by thinking of $\rho$'s and $\lambda$'s as functions on remainders, but computing these directly in $S_n$ is also legitimate). Observe in particulat that $\lambda_{[k]}^2 = e$ for any $k$. 

    (f) Deduce that the subset $D_n\subset S_n$ consisting of $\rho_k$ and $\lambda_k$ is a subgroup.

    {\bf \noindent Note: $D_4$ is another way to interpret the group in example 8.10 of symmetries of the square: keep in mind that in this example, the $\rho_k$ are the same as our $\rho_k$, but what we call $\lambda_1, \lambda_2, \lambda_3, \lambda_4,$ respectively, the book calls $\mu_2, \delta_2, \mu_1, \delta_1$, respectively (our notation is more consistent: to see what one of the $\lambda$'s is called, just apply it to the rightmost index, which in the case of $D_4$ is $4$).}
    
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{\bf 2.} Book exercises 6.17-6.19

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{\bf 3.} Book exercise 6.32

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{\bf 4.} Book exercise 6.44

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{\bf 5.} Book exercises 8.1, 8.4, 8.5 (hint: to invert a permutation in permutation notation, just flip the top and bottom rows. So in the exercise $$\tau^{-1}= \begin{pmatrix} 2&4&1&3&6&5\\1&2&3&4&5&6  \end{pmatrix}.$$ You can then put it in more standard notation by rearranging the columns (which doesn't change what the function does), so $$\tau^{-1} = \begin{pmatrix}1&2&3&4&5&6\\3&1&4&2&6&5\end{pmatrix}.$$

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{\bf 6.} Book exercises 10.1, 10.6, 10.7

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\noindent {\bf Quiz Prep questions}
  10.2-10.4, 10.20-10.24, 10.25, 10.26

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