20131021

Please Login to access more options.

Today

We'll be presenting in small groups. Please head to your respective board and start working.

Problem (Problems from the past)

We already have people ready to present these problems. If you have not yet completed them, please spend 10 minutes tops with each. Don't let yourself get bogged down with completing each of these before class, but make sure that you at least have spent 10 minutes with each (otherwise the discussion in class is pointless).

for n in (2..20):
 Zn = Integers(n)
 Un = [x for x in Zn if gcd(ZZ(x), n) == 1]  #This creates Un
 show(table(["U("+str(n)+r") has order $\varphi($"+str(n)+"$)=$"+str(len(Un))+
             ".The elements, with orders below them, are listed below."]))
 orders=[x.multiplicative_order() for x in Un] #This computes the multiplicative order of each element.
 show(table([Un,orders])) #This creates a table of elements (top row) and their orders (bottom row).

Definition (Subgroup Generated By An Element)

Let $G$ be a group and $a\in G$. Then the subgroup generated by $a$ is the set $$\langle a\rangle = \{a^n\mid n\in\mathbb{Z}\},$$ where we define $a^0=e$ and $a^{-n}=(a^{-1})^n$.

Problem 58 (The Subgroup Generated By An Element Is Actually A Subgroup)

Let $G$ be a group with $a\in G$. Show that $\langle a\rangle$, using the definition above, is a subgroup of $G$.

Problem 59 (The Intersection Of Two Subgroups Of $\mathbb{Z}$)

Let $G=\mathbb{Z}$. Because the group operation is addition, remember that $a^2$ actually means $a+a$, and $a^5=a+a+a+a+a=5a$. Beware of this issue, as $a^n$ actually means $na$ because the group operation is addition when working in $\mathbb{Z}$.

1. What is $\langle 2\rangle$? What is $\langle 3\rangle$? Convince yourself that $\langle 2\rangle\cap \langle 3\rangle = \langle 6\rangle$.
2. What is $\langle 4\rangle$? What is $\langle 6\rangle$? Find an integer $c$ so that $\langle c\rangle=\langle 4\rangle\cap \langle 6\rangle$? Prove that your result is true.
3. If $a,b\in \mathbb{Z}$, then conjecture what $c$ should equal so that $\langle c\rangle=\langle a\rangle\cap \langle b\rangle$. You don't have to prove this result (unless you'd rather prove this result than proving part 2).

Definition (The Subgroup $n\mathbb{Z}$)

For each $n\in \mathbb{N}$ we define $n\mathbb{Z}$ to be the subgroup $\langle n\rangle = \{kn\mid k\in \mathbb{Z}\}$ of the group $\mathbb{Z}$ under addition. This is just the multiples of $n$.

When we were working with simple shift permutations of the standard 26 letter alphabet, we showed that the set of all simple shift permutations, which we denoted $H=\{\phi_n\mid n\in\mathbb{Z}\}$, was actually a set that consisted of just 26 elements, namely we showed that $H=\{\phi_0,\phi_1, \ldots,\phi_{25}\}$. Each of these simple shifts can be generated using $\phi_1$, and the order of $\phi_1$ is $|\phi_1|=26$. Using the language of groups, we showed that $$H =\langle \phi_1\rangle = \{\phi_0,\phi_1, \ldots,\phi_{25}\}.$$ We showed this was true because if $n\in \mathbb{Z}$, then we could use the division algorithm to write $n=26q+r$ where $0\leq r<26$. Then we knew that $$\phi_1^{n}=\phi_1^{26q+r}=\phi_1^{26q}\circ \phi_1^r = \phi_0\circ \phi_r = \phi_r.$$ We knew that any repeated shift of 26 letters would take us back to the identity. We now generalize this problem to show that if an element of a group has order $n$, then $\langle a\rangle$ must always consist of the elements $\{e,a,a^2,\ldots, a^{n-1}\}$.

Problem 60 (Properties Of $\langle a \rangle$ When $a$ Has Finite Order)

Let $G$ be a group with $a\in G$. Suppose that the order of $a$ is $|a|=n$. Prove the following:

1. We have $\langle a\rangle = \{e,a,a^2,\ldots, a^{n-1}\}$. (You are showing two sets are equal.)
2. We have $a^i=a^j$ if and only if $i-j$ is a multiple of $n$.
3. The order of an element equals the order of the subgroup generated by that element, namely $|a|=|\langle a\rangle|$. (How can you combine 1 and 2 to get this.)

Exercise (If $a^k=e$, Then The Order Of $a$ Divides $k$)

Suppose that $a$ is a group element with order $n$. If $a^k=e$, prove that $k$ is a multiple of $n$.

Exercise (Properties Of An Element With Infinite Order)

Let $G$ be a group with $a\in G$. Suppose that the order of $a$ is infinite. Show that $a^i=a^j$ if and only if $i=j$, and state the order of $\langle a\rangle$.

The next problem asks you to compute the center $Z(G)$ of a nonabelian group, and the result is not trivial group. The automorphisms of the square gave us a group with 8 elements. This group consisted of 4 rotations and 4 reflections. Similarly, for any regular $n$-gon we can construct the automorphism group which will have order $2n$ and consist of $n$ rotations and $n$ reflections.

Definition (The Dihedral Groups $D_{n}$)

For each integer $n\geq 2$, we define the dihedral group on $n$ vertices, written $D_{n}$, to be automorphism group of the regular $n$-gon. This group consists of $n$ rotations and $n$ reflections, so has order $2n$.

Problem 61 (The Center Of A Dihedral Group)

Let $G=D_4$, the automorphism group of the square. Recall that $Z(G)$ is the center of the group, or the set of elements that commute with every element of the group.

1. What is $\langle R_{90} \rangle$? What is $\langle R_{180} \rangle$? What is $\langle R_{270} \rangle$? What is $\langle H \rangle$?
2. Does $R_{90}\in Z(G)$? Explain. (Does $R_{90}$ commute with every element in $G$? In particular, does $R_{90}H=HR_{90}$?)
3. Compute the center $Z(G)$ and show that it consists of more than just $R_0$. Make sure you can explain why each element is either in $Z(G)$, or not in $Z(G)$.

For more problems, see AllProblems