Math 120A HW 4

1. Let $(G, +)$ be an abelian group and let $a, b \in G$. (Here we follow the "additive notation" convention of denoting the operation of an arbitrary abelian group by "$+$". Accordingly, we write $0$ for the identity and write $na$ instead of $a^n$, for example in the definition of the order of $a$.)
   1. Show that if $a$ and $b$ have order $n \in \mathbb{Z}^+$ then $a+b$ has order less than or equal to $n$.
   2. Find an example of $G$, $a$, $b$, and $n$ as above such that $a$ and $b$ have order $n$, but $a+b$ has order less than $n$.
   3. Find an example of $G$, $a$, $b$, and $n$ as above such that $a$ and $b$ have order $n$ and $a+b$ also has order $n$. In this example, use $n \ne 1$ to avoid triviality; do not use the example $a = b = 0$.
2. Let $A$ be a three-element set $\{x,y,z\}$. Let $S_A$ be the group of permutations of $A$ (i.e. bijections from $A$ to $A$) under the operation of composition. So $S_A$ is a nonabelian group of order 6, as we discussed in class some time ago. Find an example of elements $f, g \in S_A$ such that $f$ and $g$ have order 2 and their composition $g \circ f$ has order 3. (This shows that Exercise 1a cannot be generalized to nonabelian groups.)
3. Let $H$ be a subgroup of $\mathbb{Z}$. Prove that $H$ is cyclic. More specifically: in the case that $H$ is nontrivial, prove that $H = \langle a \rangle$ where $a$ is the least positive element of $H$. (First, say why $H$ must have a positive element in this case.) Note 1: to prove that two subgroups of a given group (or more generally two subsets of a given set) are equal, one usually proves that each is a subset of the other. Note 2: your argument can be generalized to show that subgroups of any cyclic group are cyclic, but I won't ask you to do this now.
4. Consider $S_4$, the symmetric group on the four-element set $\{1,2,3,4\}$.
   1. Give an example of an element $\rho$ of $S_4$ of order $4$.
   2. Give an example of two elements $\sigma_1$, $\sigma_2$ of $S_4$, each of order $2$, such that $\sigma_1 \sigma_2 = \sigma_2 \sigma_1$.
   3. Give an example of two elements $\tau_1$, $\tau_2$ of $S_4$, each of order $2$, such that $\tau_1 \tau_2 \ne \tau_2 \tau_1$.
5. Calculate the orders of the following elements in the group $S_5$:
   1. $\sigma = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 4 & 2 & 1 & 5 & 3 \end{pmatrix}$
   2. $\tau = \begin{pmatrix} 1 & 2 & 3 & 4 & 5\\ 4 & 3 & 2 & 5 & 1 \end{pmatrix}$
6. Let $A$ be a set and let $x \in A$. Prove that the set $H = \{ \sigma \in S_A : \sigma(x) = x\}$ is a subgroup of the symmetric group $S_A$.
7. Recall that the dihedral group $D_4$ is the subgroup of $S_4$ corresponding to the symmetries of a regular 4-gon (square) with vertices numbered 1, 2, 3, 4 in clockwise (or counterclockwise) order. Four of the elements of $D_4$ are rotations (including the identity) and four of the elements are reflections. Give a geometric condition that says when two different reflections $\rho_1, \rho_2 \in D_4$ commute with each other (i.e. satisfy $\rho_1\rho_2 = \rho_2\rho_1$.)
8. Draw a regular pentagon and label its vertices clockwise: 1, 2, 3, 4, 5. For each vertex, draw a line through going through that vertex and through the midpoint of the opposite side. (Note: this is different than what we did with the square because the pentagon has an odd number of sides.) Pick two of these lines and write down the corresponding permutations $\rho_1, \rho_2$ of the set $\{1,2,3,4,5\}$. Calculate the product $\rho_1\rho_2$ of these permutations and show that it corresponds to a rotation. What is the angle of rotation?
9. Define the following subset of the symmetric group $S_{\mathbb{Z}}$: $$ D_\infty = \{\sigma_n : n \in \mathbb{Z}\} \cup \{\rho_n : n \in \mathbb{Z}\}, $$ where $\sigma_n$ denotes the "shift by $n$" permutation of $\mathbb{Z}$ defined by $\sigma_n(i) = i+n$ and $\rho_n$ denotes the "refect across $n/2$" permutation of $\mathbb{Z}$ defined by $\rho_n(i) = n-i$. Prove that $D_\infty$ is a subgroup of $S_{\mathbb{Z}}$. (It is called the infinite dihedral group.) Hint: for closure under multiplication there are four cases: shift times shift, shift times reflection, and so on. Here the names "shift" and "reflection" are just for intuition; you do not need to argue geometrically. It is not hard to prove directly from the definition.