##### 1

Write the following permutations in cycle notation.

\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
2 & 4 & 1 & 5 & 3
\end{pmatrix}
\end{equation*}

\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
4 & 2 & 5 & 1 & 3
\end{pmatrix}
\end{equation*}

\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
3 & 5 & 1 & 4 & 2
\end{pmatrix}
\end{equation*}

\begin{equation*}
\begin{pmatrix}
1 & 2 & 3 & 4 & 5 \\
1 & 4 & 3 & 2 & 5
\end{pmatrix}
\end{equation*}

##### 2

Compute each of the following.

\((1345)(234)\)

\((12)(1253)\)

\((143)(23)(24)\)

\((1423)(34)(56)(1324)\)

\((1254)(13)(25)\)

\((1254) (13)(25)^2\)

\((1254)^{-1} (123)(45) (1254)\)

\((1254)^2 (123)(45)\)

\((123)(45) (1254)^{-2}\)

\((1254)^{100}\)

\(|(1254)|\)

\(|(1254)^2|\)

\((12)^{-1}\)

\((12537)^{-1}\)

\([(12)(34)(12)(47)]^{-1}\)

\([(1235)(467)]^{-1}\)

##### 3

Express the following permutations as products of transpositions and identify them as even or odd.

\((14356)\)

\((156)(234)\)

\((1426)(142)\)

\((17254)(1423)(154632)\)

\((142637)\)

##### 4

Find \((a_1, a_2, \ldots, a_n)^{-1}\).

##### 5

List all of the subgroups of \(S_4\). Find each of the following sets.

\(\{ \sigma \in S_4 : \sigma(1) = 3 \}\)

\(\{ \sigma \in S_4 : \sigma(2) = 2 \}\)

\(\{ \sigma \in S_4 : \sigma(1) = 3\) and \(\sigma(2) = 2 \}\)

Are any of these sets subgroups of \(S_4\)?

##### 6

Find all of the subgroups in \(A_4\). What is the order of each subgroup?

##### 7

Find all possible orders of elements in \(S_7\) and \(A_7\).

##### 8

Show that \(A_{10}\) contains an element of order 15.

##### 9

Does \(A_8\) contain an element of order 26?

##### 10

Find an element of largest order in \(S_n\) for \(n = 3, \ldots, 10\).

##### 11

What are the possible cycle structures of elements of \(A_5\)? What about \(A_6\)?

##### 12

Let \(\sigma \in S_n\) have order \(n\). Show that for all integers \(i\) and \(j\), \(\sigma^i = \sigma^j\) if and only if \(i \equiv j \pmod{n}\).

##### 13

Let \(\sigma = \sigma_1 \cdots \sigma_m \in S_n\) be the product of disjoint cycles. Prove that the order of \(\sigma\) is the least common multiple of the lengths of the cycles \(\sigma_1, \ldots, \sigma_m\).

##### 14

Using cycle notation, list the elements in \(D_5\). What are \(r\) and \(s\)? Write every element as a product of \(r\) and \(s\).

##### 15

If the diagonals of a cube are labeled as Figure 5.26, to which motion of the cube does the permutation \((12)(34)\) correspond? What about the other permutations of the diagonals?

##### 16

Find the group of rigid motions of a tetrahedron. Show that this is the same group as \(A_4\).

##### 17

Prove that \(S_n\) is nonabelian for \(n \geq 3\).

##### 18

Show that \(A_n\) is nonabelian for \(n \geq 4\).

##### 19

Prove that \(D_n\) is nonabelian for \(n \geq 3\).

##### 20

Let \(\sigma \in S_n\) be a cycle. Prove that \(\sigma\) can be written as the product of at most \(n-1\) transpositions.

##### 21

Let \(\sigma \in S_n\). If \(\sigma\) is not a cycle, prove that \(\sigma\) can be written as the product of at most \(n-2\) transpositions.

##### 22

If \(\sigma\) can be expressed as an odd number of transpositions, show that any other product of transpositions equaling \(\sigma\) must also be odd.

##### 23

If \(\sigma\) is a cycle of odd length, prove that \(\sigma^2\) is also a cycle.

##### 24

Show that a 3-cycle is an even permutation.

##### 25

Prove that in \(A_n\) with \(n \geq 3\), any permutation is a product of cycles of length 3.

##### 26

Prove that any element in \(S_n\) can be written as a finite product of the following permutations.

\((1 2), (13), \ldots, (1n)\)

\((1 2), (23), \ldots, (n- 1,n)\)

\((12), (1 2 \ldots n )\)

##### 27

Let \(G\) be a group and define a map \(\lambda_g : G \rightarrow G\) by \(\lambda_g(a) = g a\). Prove that \(\lambda_g\) is a permutation of \(G\).

##### 28

Prove that there exist \(n!\) permutations of a set containing \(n\) elements.

##### 29

Recall that the *center* of a group \(G\) is
\begin{equation*}Z(G) = \{ g \in G : gx = xg \text{ for all } x \in G \}.\end{equation*}
Find the center of \(D_8\). What about the center of \(D_{10}\)? What is the center of \(D_n\)?

##### 30

Let \(\tau = (a_1, a_2, \ldots, a_k)\) be a cycle of length \(k\).

Prove that if \(\sigma\) is any permutation, then
\begin{equation*}\sigma \tau \sigma^{-1 } = ( \sigma(a_1), \sigma(a_2), \ldots, \sigma(a_k))\end{equation*}
is a cycle of length \(k\).

Let \(\mu\) be a cycle of length \(k\). Prove that there is a permutation \(\sigma\) such that \(\sigma \tau \sigma^{-1 } = \mu\).

##### 31

For \(\alpha\) and \(\beta\) in \(S_n\), define \(\alpha \sim \beta\) if there exists an \(\sigma \in S_n\) such that \(\sigma \alpha \sigma^{-1} = \beta\). Show that \(\sim\) is an equivalence relation on \(S_n\).

##### 32

Let \(\sigma \in S_X\). If \(\sigma^n(x) = y\), we will say that \(x \sim y\).

Show that \(\sim\) is an equivalence relation on \(X\).

If \(\sigma \in A_n\) and \(\tau \in S_n\), show that \(\tau^{-1} \sigma \tau \in A_n\).

Define the *orbit* of \(x \in X\) under \(\sigma \in S_X\) to be the set
\begin{equation*}{\mathcal O}_{x, \sigma} = \{ y : x \sim y \}.\end{equation*}
Compute the orbits of each of the following elements in \(S_5\):
\begin{align*}
\alpha & = (1254)\\
\beta & = (123)(45)\\
\gamma & = (13)(25).
\end{align*}

If \({\mathcal O}_{x, \sigma} \cap {\mathcal O}_{y, \sigma} \neq \emptyset\), prove that \({\mathcal O}_{x, \sigma} = {\mathcal O}_{y, \sigma}\). The orbits under a permutation \(\sigma\) are the equivalence classes corresponding to the equivalence relation \(\sim\).

A subgroup \(H\) of \(S_X\) is *transitive* if for every \(x, y \in X\), there exists a \(\sigma \in H\) such that \(\sigma(x) = y\). Prove that \(\langle \sigma \rangle\) is transitive if and only if \({\mathcal O}_{x, \sigma} = X\) for some \(x \in X\).

##### 33

Let \(\alpha \in S_n\) for \(n \geq 3\). If \(\alpha \beta = \beta \alpha\) for all \(\beta \in S_n\), prove that \(\alpha\) must be the identity permutation; hence, the center of \(S_n\) is the trivial subgroup.

##### 34

If \(\alpha\) is even, prove that \(\alpha^{-1}\) is also even. Does a corresponding result hold if \(\alpha\) is odd?

##### 35

Show that \(\alpha^{-1} \beta^{-1} \alpha \beta\) is even for \(\alpha, \beta \in S_n\).

##### 36

Let \(r\) and \(s\) be the elements in \(D_n\) described in Theorem 5.10.

Show that \(srs = r^{-1}\).

Show that \(r^k s = s r^{-k}\) in \(D_n\).

Prove that the order of \(r^k \in D_n\) is \(n / \gcd(k,n)\).