If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read the sections "Properties of Integers," "Modular Arithmetic" (you can skip Examples 5 and 6 on p. 9-14 if you want), and "Mathematical Induction." Here is a PDF version of Chapter 0 from Gallian; you'll need the password I distributed by email to read this document.

($\star$) Prove that for any integer $n$, the numbers $4n+1$ and $11n+3$ are relatively prime. (I.e., prove that $\gcd(4n+1,11n+3) = 1$.)
Suppose that $a$ and $b$ are nonzero integers. Prove that every element of $$S = \{ka+\ell b: k,\ell \in \mathbb{Z} \mbox{ and }ka+\ell b>0\}$$ is a multiple of $\gcd(a,b)$.
Suppose that $a,b \in \mathbb{Z}$, and that $b>0$. By the Division Algorithm, we know that there exist $q,r \in \mathbb{Z}$ so that $a = bq+r$ and $0 \leq r < b$. If $r>0$, prove that $\gcd(a,b) = \gcd(b,r).$

Here are the Solutions.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read the sections "Equivalence Relations" and "Functions (Mappings)" from Chapter 0. Read the section "Symmetries of a square" on pages 31-34, and also read the section "Definitions and Examples of Groups" on pages 42-49. (Here are PDF versions of Chapter 1 from Gallian and Chapter 2 from Gallian).

($\star$) For each of the following values of $a$ and $n$, find the remainder $r$ upon dividing $a$ by $n$. (Hint: If you do these problems in a clever way, none should be too computationally taxing.)
1. $a=110\cdot 74$ and $n=115$
2. $a=-184$ and $n=13$
3. $a=3^{20}$ and $n=24$
Use induction to prove that for any nonzero real number $x > -1$ and any integer $n \geq 2$, one has $$(1+x)^n > 1+nx.$$
Use the Euclidean algorithm to calculate $(32575,450)$. Then, use your calculations to express $(32575,450)$ as an integral linear combination of $32575$ and $450$.

Here are the Solutions.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

($\star$) Suppose that $a, b$ and $m$ are integers, and suppose further that $m>0$. Prove that if $(a,m) = 1$ and $(b,m) = 1$, then $(ab,m)=1$.
The following problems involve some basic notions from linear algebra. We will use $M_n(\mathbb{R})$ to stand for the set of all $n \times n$ matrices whose entries are real numbers, and $GL_n(\mathbb{R})$ to stand for the set of all $n \times n$ invertible matrices whose entries are real numbers. If $A \in M_n(\mathbb{R})$, then $A^T$ is the transpose of $A$.
1. We say that $A \in M_n(\mathbb{R})$ is similar to $B \in M_n(\mathbb{R})$ (written $A \sim B$) if there exists some $P \in GL_n(\mathbb{R})$ so that $PAP^{-1} = B$. Prove that similarity is an equivalence relation. (That is, show that the relation $R = \{(A,B) \in M_n(\mathbb{R}) \times M_n(\mathbb{R}): A \sim B\}$ is an equivalence relation.)
2. We say that $A \in M_2(\mathbb{R})$ is adjacent to $B \in M_2(\mathbb{R})$ (written $A \leftrightarrow B$) if there exists some $P \in M_2(\mathbb{R})$ so that $PAP^{T} = B$. Is adjacency an equivalence relation? (That is, is the relation $R = \{(A,B) \in M_2(\mathbb{R}) \times M_2(\mathbb{R}): A \leftrightarrow B\}$ an equivalence relation?) If so, prove it. If not, give a specific counterexample that shows the failure of one of the properties of an equivalence relation.
Suppose that $\alpha: A \to B$ and $\beta: B \to C$ are functions so that $\beta \circ \alpha$ is an injection. Prove that $\alpha$ is an injection. Is it necessarily the case that $\beta$ is an injection as well? If so, prove it; if not, find specific examples of functions $\alpha,\beta$ so that $\beta \circ \alpha$ is an injection, but $\beta$ is not.

Here are the Solutions.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read pages 46-53; you can skip the section labelled "Historical note" on pages 52 and 53 if you want. Also read Chapter 3.

Show that the set $\mathbb{R}$ under the operation $x \triangle y = x + y + 1$ is a group. (Note: the right hand side of this expression is the "usual" real addition.)
($\star$) Suppose you are told that $G$ is a group of six elements from $GL_2(\mathbb{R})$ whose operation is matrix multiplication. Suppose you are also told that $$\left(\begin{array}{cc}0&1\\1&0\end{array}\right) \in G \quad \mbox{ and } \quad \left(\begin{array}{rr}-1&-1\\0&1\end{array}\right) \in G.$$
1. What are the other elements in $G$? Be sure to fully justify each of your claims using the appropriate axioms and/or theorems.
2. Write out a Cayley table for $G$.
3. Is $G$ abelian?
Suppose that $(G,\bullet)$ is a group, and let $a,b \in G$ be given. Prove that for every positive integer $n$, one has $$((a\bullet b)\bullet a^{-1})^n = (a\bullet b^n)\bullet a^{-1}.$$ Your answer should have you carefully applying the axioms of a group to reach the desired conclusion. [Hint: You are trying to prove a statement for every positive integer $n$. What proof technique would be good to use in this situation? Also: for $g \in G$, we define $g^n$ as $g^n = g^{n-1} \bullet g$.]

Here are the Solutions to this assignment.

For this week's assignment, make sure that you very carefully apply the appropriate group axioms when solving problems. All claims should be supported by careful, detailed arguments. This is your chance to show off that your precision!

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Re-read chapter 3 and scan chapter 4.

Suppose that $G$ is a group (whose operation is written multiplicatively), and which satisfies the property that for all $g \in G$ one has $g^2 = e$. (Here and throughout the course, $g^2$ stands for the element $gg$.) Prove that $G$ is abelian. [Hint: think about the element $ab$.]
($\star$) Suppose that $G$ is a group with the following property: if $a,b \in G$ and $n$ is a positive integer, then $(ab)^n = a^nb^n$. Prove that $G$ is abelian.
Consider the set of matrices $$G = \left\{\left(\begin{array}{ccc}1&a&b\\0&1&c\\0&0&1\end{array}\right): a,b,c \in \mathbb{R}\right\}.$$ Prove that under the operation of matrix multiplication, $G$ is a group. [Hint: In class we saw that matrix multiplication is associative, and hence $G$ "inherits" associativity from $\mathcal{M}_3(\mathbb{R})$. So don't worry about proving associativity.]

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Reread chapter 4 and read chapter 5.

Suppose that $G$ is a group and $g \in G$. Prove that $|g| = |g^{-1}|$.
Suppose that $G$ is a group and $g \in G$. Suppose you also know that $g^2 \neq e$ and $g^5 \neq e$, but that $g^{10}=e$. Prove that $|g| = 10$. (Sorry, but for this problem you cannot use the result from chapter 4 that says that if $g^k = e$, then $|g| \mid k$.)
($\star$) Suppose that $G$ is a group, and you are told that $a,b \in G$ satisfy $|a| = 4$ and $|b| = 2$. If $a^3b = ba$, what is the order of $ab$?
Suppose that $G$ is a group, and you are told that $a,b \in G$ satisfy $|a| = 4$ and $|b| = 2$. Explain why $a^2b = ba$ is impossible.

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 5.

($\star$)
1. Let $G$ be an abelian group, and let $n$ be a positive integer. Define $$H = \{g \in G: g^n = e\}.$$ Prove that $H\leq G$. [Hint: you may want to prove that for any $a,b \in G$ and $k$ a positive integer, we have $(ab)^k = a^kb^k$.]
2. Give a specific example of a group $G$ and a positive integer $n$ so that $$\{g \in G: g^n = e\} \not\leq G.$$ Be sure to explicitly exhibit the failure of one of the group axioms to justify your claim.
On your third homework assignment, you considered $G = \left\{I,A,B,AB,BA,ABA\right\} \subseteq GL_2(\mathbb{R})$ which had the following Cayley table: $$ \begin{array}{c|cccccc} &I&A&B&AB&BA&ABA\\ \hline I&I&A&B&AB&BA&ABA \\ A&A&I&AB&B&ABA&BA \\ B&B&BA&I&ABA&A&AB \\ AB&AB&ABA&A&BA&I&B \\ BA&BA&B&ABA&I&AB&A \\ ABA&ABA&AB&BA&A&B&I \\ \end{array} $$
1. Compute $Z(G)$.
2. For each $g \in G$, compute $|g|$. Make sure you explain your conclusions.
3. Compute $C(A)$ and $C(AB)$.
Suppose that $G$ is a group, $g \in G$ is given, and $H$ is a subgroup of $G$. Prove that $$gHg^{-1} = \{ghg^{-1}: h \in H\}$$ is a subgroup of $G$. (Such a subgroup is called a conjugate of $H$.)

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 6.

Let $a,b \in G$, and suppose you know that $|a| = 10$ and $|b| = 77$. Prove that $\langle a \rangle \cap \langle b \rangle = \{e\}$. [Note: this question deals with the intersection of two sets. In general, if $A$ and $B$ are sets within a larger set $X$, then $A \cap B$ is the set of all $x \in X$ which satisfy both $x \in A$ and $x \in B$.]
($\star$) Suppose that $G = \langle a \rangle$, and that $|a| = 24$.
1. Find (with proof) all generators of $G$.
2. Give a subgroup of $G$ with $6$ elements. Find all generators of this subgroup.
Let $n$ be a positive integer that is at least $3$. Show that $U(2^n)$ has at least three elements of order $2$. Then explain why this proves $U(2^n)$ is not cyclic. [Hint for the first part: For a few small values of $n$, find a few $g \in U(2^n)$ which satisfy $g^2 = 1$. Use these cases to make a conjecture about values in an arbitrary $U(2^n)$ which have order $2$, and then prove your conjecture.]

Here are the Solutions to this assignment.

Read chapter 7.

Do problem (5) from the the permutation handout from class. (If you want to use information about $|\gamma|$ in your calculation, be sure to justify any claim you make concerning the value of $|\gamma|$ either with direct computations or an appropriately cited theorem.)
Do problem (6) from the the permutation handout from class. (This is essentially a calculation, so requires no justification.)
($\star$) Do problem (8) from the the permutation handout from class.
Do problem (9) from the the permutation handout from class.
($\star$) Do problem (11) from the the permutation handout from class.
Suppose that $G$ is a group and $a \in G$. Suppose further that $|a|=5$. Prove that $C(a) = C(a^3)$. [Note: this question asks you to prove that two sets are equal. In general, to show that two sets $A$ and $B$ satisfy $A = B$, you have to show that $A \subseteq B$, and that $B \subseteq A$. To show that $A \subseteq B$, one has to prove that a given $a \in A$ also satisfies $a \in B$.]

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 8.

Let $i \in \{1,2,\cdots,n\}$ be given, and define $\textrm{stab}(i) = \{\sigma \in S_n : \sigma \mbox{ fixes }i\}.$ Prove that $\textrm{stab}(i)$ is a subgroup of $S_n$. (Note: this subgroup is called the stabilizer of $i$ in $S_n$. You might find it convenient to use the notation ``$\sigma[k]$'' to denote the output under the function $\sigma$ that corresponds to the input $k \in \{1,2,\cdots,n\}$. With this notation, $\text{stab}(i)$ becomes $\{\sigma \in S_n: \sigma[i]=i\}$.)
Show that an $m$-cycle is even if and only if $m$ is odd.
($\star$) Show that if $\sigma \in S_n$ has odd order, then $\sigma \in A_n$. Then, find specific elements $\alpha,\beta \in S_6$ so that $\alpha$ and $\beta$ both have even order, yet $\alpha \in A_6$ but $\beta \not\in A_6$.

Here are the Solutions to this assignment.

This assignment is NOT to be submitted for a grade. It is provided to give you practice with the concepts at the foundation of the theory of isomorphisms and homomorphisms (from lectures 14 and 15).

Reread chapters 1-6.

Prove that "isomorphic to" gives an equivalence relation on the set of groups.
Recall that for an element $a \in G$, we define $\varphi_a \in \text{Inn}(G)$ by $\varphi_a(x) = axa^{-1}$. Prove that $\varphi_h = \varphi_g$ if and only if $g^{-1}h \in Z(G)$.
In class we claimed that for any $g \in G$, we have $\varphi_g$ is an isomorphism from $G$ to itself. Prove this claim.
Suppose that $\phi:D_4 \to D_4$ is an isomorphism that satisfies $\phi(R_{90}) = R_{90}$ and $\phi(H) = V$. Prove that $\phi$ must be the inner automorphism defined by $R_{90}$. [Hint: You need to show that for each $g \in D_4$, we have $\phi(g) = \varphi_{R_{90}}(g)$.]

Here are the Solutions to this assignment.

The following are OPTIONAL problems you may submit for bonus credit on the first exam. If you choose to work on these problems, you must do so BY YOURSELF. You cannot consult classmates, tutors in the helproom, friends, internet resources, other texts, etc. The only resources you may consult are your notes, your textbook, and the professor. Partial credit will be given on these problems only if you've made substantial progress towards a final solution. Please make sure your work is neat and well presented; answers without supporting work and explanation will receive no credit.

You can get a maximum of 5 bonus points on the first exam; you get 2 bonus points for attending either of the math department colloquia on September 24 or October 6 (hence a maximum of 4 points for attending colloquia). If you get more than 5 bonus points (combining points from these problems and attendance at colloquia), additional points can be carried over to the second midterm.

You should not spend time working on these problems if you are not already prepared to take the exam. These problems are not necessarily representative of exam problems, and are by no means a substitute for the preparation you get by reviewing your class notes, reading posted solutions to quiz and homework problems, and working on a variety of problems from the book.

($\star$) Suppose that $G$ is a group that has exactly three subgroups (including $\{e\}$ and $G$). Prove that $G$ is cyclic and $|G|=p^2$ where $p$ is prime.
($\star$) If $\beta \in S_7$ satisfies $|\beta^3|=7$, prove that $|\beta|=7$. Find an element in $\gamma \in S_{10}$ so that $|\gamma^3|=7$ but so that $|\gamma|\neq 7$.
($\star$) Suppose that $G$ is a group with even order. Prove that $G$ has at least one element of order $2$.
($\star$) Suppose that $H$ is a subgroup of $S_n$. Prove that either $H$ has no odd elements, or that precisely half the elements of $H$ are odd.
($\star$) Compute, with proof, the number of elements of order $3$ in $S_7$.
($\star$) Find an explicit example of a group $G$ and an element $a \in G$ so that $1<|\phi_a| < |a|$.
($\star$) [2 pts] Suppose that $Q$ and $N$ are groups (with operations $*$ and $\star$, respectively) and that there is a homomorphism $\theta: Q \to \textrm{Aut}(N)$; we'll use the notation $\theta_q$ to stand for the automorphism of $N$ which $q$ maps to under $\theta$ (hence for each $q \in Q$ there exists a $\theta_q \in \textrm{Aut}(N)$). Now we'll define $N \rtimes_\theta Q$ to be the set $N \times Q$ equipped with operation $$(n_1,q_1)(n_2,q_2) = (n_1 \star \theta_{q_1}(n_2),q_1 * q_2).$$ Prove that this is a bona fide binary operation on $N \rtimes_\theta Q$, and that $N \rtimes_\theta Q$ forms a group under this binary operation. [This group is called the semi-direct product of $N$ and $Q$ relative to $\theta$.]

Reread chapters 7-8.

Prove that $U(8)$ is not isomorphic to $U(10)$.
($\star$) Prove that the function $f:\mathbb{Z}_3 \to \mathbb{Z}_6$ given by $f(x) = 3x \pmod{6}$ is not a homomorphism.
Prove that the determinant map is a homomorphism from $\text{GL}_n(\mathbb{R})$ to $\mathbb{R}^\times$. What is its kernel? (Note: You can use familiar properties about matrices and determinants that you learned in linear algebra.)

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $G \simeq Q$. Prove that $\text{Aut}(G) \simeq \text{Aut}(Q)$.
Prove that a group $G$ is abelian if and only if the function $f(g) = g^{-1}$ is a homomorphism.
($\star$) Let $H \leq G$ and $x \in G$. Prove that $H \simeq xHx^{-1}$.

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Prove that $\mathbb{R}^\times \oplus \mathbb{R}^\times \not\simeq \mathbb{C}^\times$.
($\star$) Suppose $H_1 \leq G_1$ and $H_2 \leq G_2$, where $(G_1,\star)$ and $(G_2,\bullet)$ are groups. Prove that $H_1 \oplus H_2 \leq G_1 \oplus G_2$.
Compute the number of elements of order $15$ in $S_3 \oplus \mathbb{Z}_{30} \oplus U(9)$.

You might also want to try to prove the following result. (It will give you good practice with some of the concepts we've been discussing recently.) If you do this problem, DO NOT submit it with your pset solutions.

Suppose that $n \in \mathbb{N}$, and let $\ell \in U(n)$ be given. Prove that the function $f:\mathbb{Z}_n \to \mathbb{Z}_n$ given by $f(x) = \ell x \pmod{n}$ is an isomorphism.

It's worth observing that this problem shows that the function $\psi:\text{Aut}(\mathbb{Z}_n) \to U(n)$ we constructed in class is indeed surjective, since the function $f$ above has $\psi(f) = f(1) = \ell$. Hence this completes our proof that $\text{Aut}(\mathbb{Z}_n) \simeq U(n)$.)

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $\phi:G \to Q_8$ is a surjective homomorphism with $|\ker(\phi)|=6$.
1. Prove that $G$ has at least one subgroup of size $12$, and at least three (distinct) subgroups of order $24$. [Hint: what do you know about subgroups of $Q_8$?]
2. Can you make any conclusions about whether $G$ is abelian or not? Justify your answer.
3. Suppose we drop the assumption that $\phi$ is surjective. Give an explicit example of a group $G$ and a homomorphism $\phi:G \to Q_8$ so that $|\ker(\phi)|=6$, and yet $G$ does not contain any subgroups of order $12$ or $24$.
($\star$) Is it always true that $\text{Aut}(G \oplus H) \simeq \text{Aut}(G) \oplus \text{Aut}(H)$? Prove or give a specific counterexample. [Hint: to get some intuition going, you might try thinking about some small groups $G$ and $H$ to see if this statement holds.]
Find some $n \neq 20$ so that $U(n) \simeq U(20)$. Justify your answer.

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

($\star$) In class we defined a Carmichael number to be a composite integer $n$ with the property that for all $b \in U(n)$ we have $b^{n-1} = 1 \pmod{n}$. Prove $n=561$ is a Carmichael number. [Hint: what group is $U(561)$ isomorphic to?]
For each of the following groups $G$ and subgroups $H$, list all the left cosets of $H$ in $G$, and then all the right cosets of $H$ in $G$. (In your solutions, be sure to distinguish where you're listing left cosets versus where you're listing right cosets.)
1. $H = \langle (1234)\rangle$ and $G = S_4$
2. $H = \langle (123) \rangle$ and $G = A_4$
Suppose that $|G| = 55$. Prove that $G$ must have an element of order $5$, and that it must have an element of order $11$. [Hint: what can you say about orders of elements in $G$, and how many of a given order could exist in $G$?]

Here are the Solutions to this assignment.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 9 (you can skip the section "Internal Direct Products") and the section of Chapter 10 entitled "The First Isomorphism Theorem."

Find an explicit group $G$, subgroup $K$, and elements $a,b \in G$ so that $aK = bK$ but $ab^{-1} \not\in K$. Why does this result not contradict the properties of cosets we discussed in class?
($\star$) Find two proper subgroups $K$ of $D_4$ (i.e., subgroups other than $\{e\}$ and $D_4$) such that for all $a \in D_4$ we have $aK = Ka$.
Let $\displaystyle H = \left\{\left(\begin{array}{cc}a&b\\0&c\end{array}\right): a,b,c \in \mathbb{R} \mbox{ and }ac \neq 0\right\}.$ Is $H \lhd \textrm{GL}_2(\mathbb{R})$?Justify your assertion.

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 11.

1. What is the order of the element $3\langle 16\rangle$ within the group $U(35)/\langle 16 \rangle$?
2. Suppose that $G$ is a group and $H \lhd G$; suppose that $g \in G$ is given so that $g$ has finite order. Prove that the order of $gH$ (as an element of $G/H$) is finite and divides the order of $g$ (as an element of $G$). [Note: we now have two meanings for "the order of $gH$;" one is in thinking of gH as a set of elements, and the other in thinking of gH as an element of the group $G/H$. In both parts of this problem, we're interested in the order of these cosets as elements in their respective factor groups.]
Suppose that $N \lhd G$ and $H \leq G$, and define $NH = \{nh: n \in N, h \in H\}$. Prove that $NH \leq G$.
Suppose that $|G| = p^3$ and that $Z(G) \neq \{e\}$ and $Z(G) \neq G$. Prove that $Z(G) \approx \mathbb{Z}_p$.

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 12.

Suppose that $f:G \to Q$ is a homomorphism and that $H \lhd G$ which satisfies $H \subseteq \ker(f)$. Prove that $$\tilde f: G/H \to Q$$ defined by $\tilde f\left(aH\right) = f(a)$ is well-defined and a homomorphism. [Note: you can describe this phenomenon in a couple of ways. Some people say $\tilde f$ is the homomorphism on $G/H$ induced by $f$; others say that $f$ descends to the homomorphism $\tilde f$ on $G/H$.]
Suppose that $G$ is a group so that $[G:Z(G)] \leq 3$. Prove that $G$ is abelian.
Prove that a non-trivial, finite abelian group $G$ is simple if and only if $G \simeq \mathbb{Z}_p$ for some prime $p$. (Recall that a group $G$ is called simple if the only normal subgroups it contains are $\{e_G\}$ and $G$.) [Hint: Cauchy's theorem on finite abelian groups will be useful.]

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapter 13.

Prove the second isomorphism theorem: If $K \leq G$ and $N \lhd G$, then $$K/(K\cap N) \approx KN/N.$$ (Note: The set $KN$ is defined as $$KN = \{kn : k \in K \mbox{ and }n \in N\}.$$ This is much like the subgroup $NK$ from the last assignment, so don't bother with showing that $KN$ is a subgroup. For this statement to make sense, though, you should check that $N \lhd KN$.)

[Hint: Use the first isomorphism theorem.]
Prove the third isomorphism theorem: if $H,K$ are normal subgroups of $G$ with $H \leq K \leq G$, then $$\frac{G/H}{K/H} \approx G/K.$$[Hint: Use the first isomorphism theorem.]
Suppose that $k \mid n$. Prove that $\mathbb{Z}_n/\langle k \rangle \approx \mathbb{Z}_k$. [Note: On the left side of the isomorphism, the element $k$ is in $\mathbb{Z}_n$, and so $\langle k \rangle \subseteq \mathbb{Z}_n$ (as opposed to $k \in \mathbb{Z}$ and $\langle k \rangle \subseteq \mathbb{Z})$.]
[Hint: You could prove this with the first isomorphism theorem, but another approach begins by proving that the quotient of a cyclic group is cyclic, and then considering orders. Also: to make sure your proof is valid, be sure you understand how you're using the hypothesis $k \mid n$.]

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $G$ is an abelian group of order $2^n$. Determine (with proof) how many elements of order $2$ are in $G$; your answer should be expressed in terms of the number of summands in the primary decomposition of $G$.
For a finite group $G$, the exponent of $G$ is defined to be the smallest positive integer $k$ so that for all $g \in G$ we have $g^k = e$. For each of the possible (isomorphism classes of) abelian groups of order $100$, compute the exponent of the group. Make sure to prove your claims.
List (with proof) all the (isomorphism classes of) abelian groups of order $42917875$.

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If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Suppose that $R$ is a ring. Let $U(R)$ be the set of units of $R$, and let $ZD(R)$ be the set of zero divisors of $R$. Prove that $U(R) \cap ZD(R) = \emptyset$.
Let $F(\mathbb{R},\mathbb{R})$ be the set of all functions with domain and codomain $\mathbb{R}$. (For example, the function $f: \mathbb{R} \to \mathbb{R}$ defined by $f(x) = x^2$ is an element of $F(\mathbb{R},\mathbb{R})$.)
We'll define addition on $F(\mathbb{R},\mathbb{R})$ by letting $f+g$ be the function given by $(f+g)(x) = f(x) + g(x)$, and we'll define multiplication by letting $fg$ be the function given by $(fg)(x) = f(x)g(x)$. With operations defined in this way, $F(\mathbb{R},\mathbb{R})$ is a ring. (You don't have to prove this fact.)
1. What is the additive identity in $F(\mathbb{R},\mathbb{R})$? What is the multiplicative identity in $F(\mathbb{R},\mathbb{R})$? (Include justifications for your responses.)
2. Characterize the functions in $U(F(\mathbb{R},\mathbb{R}))$ (i.e., what is the precise property an element from $F(\mathbb{R},\mathbb{R})$ must satisfy in order for it to be invertible?). Prove that every nonzero element in $F(\mathbb{R},\mathbb{R})$ is either a unit or a zero divisor.
The ring $\mathbb{Z}_3[i]$ is the set $\{a+bi: a,b \in \mathbb{Z}_3\}$ subject to the operations $$(a+bi)+(c+di) = (a+c)+(b+d)i$$ $$(a+bi)(c+di) = (ac-bd)+(ad+bc)i.$$ The zero element in this ring is $0+0i$, and the multiplicative identity is $1+0i$. [You don't have to check these facts.]
1. It is a fact (that you don't have to check) that the eight nonzero elements of $\mathbb{Z}_3[i]$ form an abelian group under multiplication. By the Fundamental Theorem of Finite Abelian groups, it follows that the set of nonzero elements in $\mathbb{Z}_3[i]$ under multiplication is isomorphic to one of $\mathbb{Z}_2 \oplus \mathbb{Z}_2 \oplus \mathbb{Z}_2, \mathbb{Z}_4 \oplus \mathbb{Z}_2$, or $\mathbb{Z}_8$. Which is it? Justify your response. [Hint: You might find Table 13.1 on page 243 useful.]
2. For each of the eight nonzero elements of $\mathbb{Z}_3[i]$, find the multiplicative inverse. [Hint: You still might find Table 13.1 on page 243 useful.]

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This assignment is not to be turned in. Instead, work on these problems to get practice with some of the topics we discussed on Lecture 32.

Suppose that $R$ is a ring and $a \in R$ is given. Define the left-annihilator of $a$ to be $$\text{ann}_L(a) = \{r \in R: ra = 0\}.$$ Prove that $\text{ann}_L(a)$ is a subring of $R$.
Prove or find a counterexample: if $a,b$ are elements of a ring with $ab= 0$, then $ba = 0$.
Suppose for a ring $R$ there exists some positive integer $n$ with $x^n = x$ for all $x \in R$. Prove that if for some $a \in R$ we have a positive integer $m$ with $a^m = 0$, then $a=0$.
Suppose that $R$ is an integral domain. Let $S = \{(a,b): a,b \in R, b \neq 0\}$. We will say that $(a,b) \equiv (c,d)$ if $ad = bc$.
1. Prove that the above relation gives an equivalence relation.
2. Define the set $FF(R)$ to be the set of equivalence classes of elements in $S$; that is, $$FF(R) = \{[(a,b)]: (a,b) \in S\}.$$ Define operations on $FF(R)$ by $$[(a,b)]+[(c,d)] = [(ad+bc,bd)]$$ $$[(a,b)][(c,d)]= [(ac,bd)].$$ Prove that $FF(R)$ is a field under these operation. [This field is called the field of fractions of $R$. You can think of an element $[(a,b)] \in FF(R)$ as analogous to the "fraction" $\frac{a}{b}$; the fact that we need to take equivalence classes within $S$ is because we might have fractions that aren't in "lowest terms."]

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The following are OPTIONAL problems you may submit for bonus credit on the second exam. If you choose to work on these problems, you must do so BY YOURSELF. You cannot consult classmates, tutors in the helproom, friends, internet resources, other texts, etc. The only resources you may consult are your notes, your textbook, problems you've done on old homeworks, and the professor. Partial credit will be given on these problems only if you've made substantial progress towards a final solution. Please make sure your work is neat and well presented; answers without supporting work and explanation will receive no credit.

You can get a maximum of 5 bonus points on the first exam; you get 2 bonus points for attending either of the math department colloquia on October 20, November 2 or November 17.

You should not spend time working on these problems if you are not already prepared to take the exam. These problems are not necessarily representative of exam problems, and are by no means a substitute for the preparation you get by reviewing your class notes, reading posted solutions to quiz and homework problems, and working on a variety of problems from the book.

($\star$) Suppose that $A$ is a set and that $G$ is a subgroup of the permutation group $S_A$. For an element $a \in A$, we define $$\mathcal{O}_G(a) = \{g(a): g \in G\}.$$ (This set is called the orbit of $a$ under $G$. Note that $\mathcal{O}_G(a) \subseteq A$.) Prove that $\textrm{stab}(a) = \{g \in G: g(a) = a\}$ is a subgroup of $G$. Show that there is a bijection between $\mathcal{O}_G(a)$ and left cosets of $\textrm{stab}(a)$. Conclude that $$|G| = \left|\textrm{stab}(a)\right|\left|\mathcal{O}_G(a)\right|.$$ (This is often called the orbit-stabilizer theorem.)
($\star$) Suppose that $R_1,R_2,\cdots,R_n$ are rings, each with identity. Prove that $U(R_1 \oplus \cdots \oplus R_n) = U(R_1) \oplus \cdots \oplus U(R_n)$. [Here we use $U(R)$ to denote the units in the ring.]
($\star$) Suppose that $N \leq G$. Prove that $N \lhd G$ if and only if the following statement holds: for all $a,b \in G$, it follows that $ab^{-1} \in N$ if and only if $a^{-1}b \in N$.
($\star$) Let $G$ be a group and $g \in G$ be given. The conjugacy class of $g$ is the set $$[g] = \{xgx^{-1}: x \in G\}.$$ (The conjugacy classes are simply the equivalence classes given by the relation $a \sim b$ if and only if there exists some $x \in G$ so that $a = xbx^{-1}$.) Note that $[g] = \{g\}$ if and only if $g \in Z(G)$. (You don't have to prove this.)
Now suppose that $|G| = p^k$ for some prime $p$ and some positive integer $k$ . Prove that if $g \not\in Z(G)$, then $|[g]|$ is a power of $p$. [Hint: Use the orbit-stabilizer theorem above.] Use this to prove that $|Z(G)| \neq 1$ for such a group $G$.
($\star$) Prove that if $|G| = p^2$, then $G \simeq \mathbb{Z}_{p^2}$ or $G \simeq \mathbb{Z}_p \oplus \mathbb{Z}_p$. [Hint: Use the previous result.]
($\star$) Suppose that $H_1,H_2$ are index two inside of a group $G$. It is a fact (that you don't have to prove) that $H_1 \cap H_2$ is a subgroup. Prove that $H_1 \cap H_2 \lhd G$, that $H_1 \cap H_2$ is index $4$, and that $G/(H_1 \cap H_2)$ is not cyclic.
($\star$) If $G$ is a group, then if $a,b \in G$, we define the commutator $[a,b]$ of $a$ and $b$ to be the element $$[a,b]=aba^{-1}b^{-1}.$$ The commutator subgroup $[G,G]$ is the subgroup generated by commutators. This means that $x \in [G,G]$ if and only if there exists some $n \in \mathbb{N}$ and some group elements $a_1,b_1,a_2,b_2,\cdots,a_n,b_n \in G$ with $$x = [a_1,b_1][a_2,b_2]\cdots[a_n,b_n].$$ It is a fact (that you don't have to prove) that $[G,G] \leq G$. Prove that $[G,G] \lhd G$, and that $G/[G,G]$ is abelian.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

Read chapters 14 and 15.

Suppose that $a$ and $b$ are elements of some commutative ring $R$. If $a \in U(R)$ and $b^2 = 0$, prove that $a+b \in U(R)$.
Suppose that $I$ is an ideal of $\mathbb{Z}$. Prove that there exists an integer $d$ so that $I = \langle d \rangle$. [Hint: This is easy if $I = \{0\}$. If $I \neq \{0\}$, argue that $I$ contains a least positive element.]
Suppose that $R$ is a commutative ring, and $r \in R$ is given. The annihilator of $r$ --- written $\textrm{ann}(r)$ --- is the set $$\textrm{ann}(r) = \{x \in R: xr = 0\}.$$ Prove that $\textrm{ann}(r)$ is an ideal of $R$.

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The problems in this problem set are optional. You can choose to submit them for a grade or not. If you choose not to submit them for a grade, your homework average will be computed based on your other homework assignments. If you do choose to submit these problems for a grade, the grade you receive will only be used in computing your homework average if your score improves your overall homework average. The material covered in this assignment is relevant for the final exam.

If you collaborate with anyone on this problem set, include this information at the top of your submission (write something like ``I collaborated with..."). Problems marked "($\star$)" should be completed by yourself, or with assistance from the instructor.

We say that an integral domain $R$ is a Euclidean domain if there is a measure function $d:R\setminus\{0_R\} \to \mathbb{N} \cup \{0\}$ which satisfies (1) for all nonzero $a,b \in R$ we have $d(a) \leq d(ab)$ and (2) if $a,b \in R$ and $b \neq 0$, there exists $q,r \in R$ so that $a=bq+r$ and either $r=0$ or $d(r) {<} d(b)$. The function $d$ above is sometimes called a measure on $R$.
Prove that every Euclidean domain is a principle ideal domain. In other words: prove that if $R$ is a Euclidean domain and $I$ is an ideal of $R$, then there exists some $d \in R$ with the property that $\langle d \rangle = I$. [Hint: Use problem 2 from Assignment 24 as inspiration, but replace ``least positive element in $I$'' with ``element with minimal measure in $I$''.]
Let $\mathbb{F}$ be a field, and consider the ring of polynomials $\mathbb{F}[x]$ with coefficients in $\mathbb{F}$. (To be explicit, as a set we have $$\mathbb{F}[x] = \{a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x+a_0: n \in \mathbb{N} \quad \mbox{ and } \quad a_0,\cdots,a_n \in \mathbb{F}\},$$ and addition and multiplication are done in the usual ways: if $f,g \in \mathbb{F}[x]$ are given and we write $a_i$ for the coefficient of $x^i$ in the expression for $f$ and $b_i$ for the coefficient of $x^i$ in the expression for $g$, then the coefficient of $x^i$ in $f+g$ is $a_i+b_i$, and the coefficient of $x^i$ in $fg$ is $\sum_{j=0}^i a_jb_{i-j}$.)
1. Argue that the degree function $\deg:\mathbb{F}[x] \to \mathbb{N}\cup \{0\}$ makes $\mathbb{F}[x]$ into a Euclidean domain. [Recall that if $f$ is a polynomial of the form $f = a_nx^n + a_{n-1}x^{n-1} + \cdots + a_1x + a_0$ and $a_n \neq 0$, then $\deg(f) = n$.]
2. Find a specific example which shows that the degree function on $\mathbb{Z}[x]$ does not satisfy the required properties to be a measure on $\mathbb{Z}[x]$. (Note: one could argue that the degree function is not a measure far more abstractly, as follows: if it were a measure then $\mathbb{Z}[x]$ would be a Euclidean domain, and (by the previous problem) therefore a PID. But we stated in class that $\langle x,2 \rangle$ is not a principle ideal, and so $\mathbb{Z}[x]$ is not a PID. This argument has the benefit of not only showing that the degree function isn't a measure, but that NO function is a measure on $\mathbb{Z}[x]$.)
Recall that $\mathbb{Z}_2[x]$ is the set of polynomials with coefficients in $\mathbb{Z}_2$. (So, for instance, $x+1 = x-1$ in $\mathbb{Z}_2[x]$, since $2=0$ in $\mathbb{Z}_2$.) Since $\mathbb{Z}_2$ is a field, the previous problem gives that the degree function is a measure on $\mathbb{Z}_2[x]$.
1. Prove that $\mathbb{Z}_2[x]/\langle x^3+x+1\rangle$ has eight elements. [Hint: argue that every element in the quotient is represented by a polynomial of "small" degree.]
2. Compute the multiplicative inverse of every nonzero element. [Hint: if you can argue that $(f(x)+\langle x^3+x+1\rangle)(g(x)+\langle x^3+x+1\rangle) = 1+\langle x^3+x+1\rangle$, then you'll have shown that $f(x) = g(x)^{-1}$ and that $g(x) = f(x)^{-1}$.]

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