MATH-UA 343 Algebra – Problem Sets

1. Read Chapters 1, 2, 3.1-3.2 of Judson
2. Judson, 1.3, p. 17: #6, 8, 19, 22abc

3. Let \(f\from A\to B\), \(g\from B\to A\). We say that \(g\) is the inverse of \(f\) if \(f\circ g=\id_B\) and \(g\circ f=\id_A\).

   Is it possible for a function \(f\from A\to B\) to have more than one inverse? If so, give an example. If not, prove that it's impossible by showing that if \(g\) and \(h\) are both inverses of \(f\), then \(g=h\).

4. If you need practice with induction, do some of exercises 1-14 in Section 2.3. You don't need to turn these in.
5. Judson, 3.4, p. 48: #3, 13

1. Read Chapters 3.3, 4.1-4.2 of Judson
2. Let \(x_1,\dots,x_n\in G\). Show by induction that \[(x_1x_2\dots x_n)^{-1}=x_n^{-1}x_{n-1}^{-1}\dots x_1^{-1}.\]
3. Let \(f\from A\to B\), \(g\from B\to A\). We say that \(g\) is a left inverse of \(f\) if \(g\circ f=\id_A\). Show that \(f\) has a left inverse if and only if it is injective. Find an example of a map \(f\from \N\to \N\) with infinitely many left inverses.
4. Judson, 3.4, p. 48: 45, 46
5. Let \(D_4\) be the symmetry group of the square. Let \(\sigma\in D_4\) be rotation by 90 degrees counterclockwise and let \(\tau\in D_4\) be reflection across a horizontal axis. How many elements does \(D_4\) have? Show that every element can be written in the form \(\tau^m \sigma^n\) for some \(m\) and \(n\). Is \(D_4\) abelian?
6. List the subgroups of \(\Z_{20}\) and the generators of \(\Z_{20}\). What is the relationship between the two lists?

1. Read Chapter 5 of Judson.
2. Study for the quiz.
3. Judson, 4.4, p. 66: 11, 23
4. Let \(n>0\) be an integer. Let \(H\) be a nontrivial subgroup of \(\Z_n\) and let \(k\) be the smallest nonzero element of \(H\). Show that \(k\) is a divisor of \(n\) and that \(|H|=\frac{n}{k}\).
5. Show that if \(G\) has no nontrivial proper subgroups, then \(G\) is either the group of order \(1\) or a cyclic group whose order is prime.
6. Do parts of exercises 1 and 2 of section 5.3 to familiarize yourself with cycle notation and multiplying permutations. Do not turn this in.
7. Let \(\sigma=(123)(45)\in S_5\). What is the order of \(\sigma\) and what are the powers of \(\sigma\)? What is the order of \(\tau=(12345)(6789)\)?

1. Read Judson, Chapter 6.1
2. Judson, 5.3, p. 84: 13
3. Is \(\{\sigma\in S_5 \mid \sigma(3)=3\}\) a subgroup of \(S_5\)? Is \(\{\sigma\in S_5 \mid \sigma(2)=4\}\) a subgroup of \(S_5\)? Why or why not?
4. Let \(\sigma\in S_n\) be a cycle of length \(k\). Show that \(\sigma\) can be written as a product of \(k-1\) transpositions. When is \(\sigma\) an even permutation? When is \(\sigma\) an odd permutation?
5. Judson, 5.3, p. 84: 21
6. Suppose that \(\sigma=c_1\dots c_n\), where \(c_i\) is a cycle of length \(k_i\). How can you determine whether \(\sigma\) is odd or even from the \(k_i\)'s?

7. Let \(\sigma=(12)\) and let \(\tau=(12345)\). Write the following as products of \(\sigma\) and \(\tau\). (You can use \(\sigma\) and \(\tau\) any number of times in each product.)

   1. \((54321)\)
   2. \((23)\)
   3. \((13)\)

   In fact, any element of \(S_5\) can be written as a product of \(\sigma\) and \(\tau\)!

1. Study for the quiz in recitation on Friday, March 8
2. Read Judson, Chapter 6.2-6.3
3. Judson, Chapter 5.3 (Permutation Groups), p. 84: 30
4. Judson, Chapter 6.4 (Cosets and Lagrange's Theorem), p. 94: 1, 6, 11, 13
5. Let \(G=S_n\) and let \(H=\{\sigma\in S_n \mid \sigma(1)=1\}\) so that \(H\) is a subgroup of \(G\). Show that the left cosets of \(H\) are the sets \(\{\sigma\in S_n \mid \sigma(1)=k\}\) for \(k=1,\dots,n\). Verify that \(|G|=[G:H]\cdot |H|\). What are the right cosets of \(H\)?

1. Study for the quiz in recitation on Friday, March 8
2. Read Judson, Chapter 9.1
3. Judson, Chapter 6.4 (Cosets and Lagrange's Theorem), p. 94: 8, 9
4. Let \(H\) be a subgroup of \(G\) and let \(C\) be a left coset of \(H\). Show that for any \(g\in G\), the set \(gC=\{gc\mid c\in C\}\) is also a left coset of \(H\).
5. Suppose that \(H\) is a subgroup of \(G\) of index \(k\) and let \(C_1,\dots, C_k\) be the left cosets of \(H\). Let \(g\in G\). Show that there is a permutation \(\sigma\in S_k\) such that \(gC_i=C_{\sigma(i)}\). (Alternatively, let \(L\) be the set of left cosets of \(H\). Show that the map \(f\from L\to L\) such that \(f(C)=gC\) is a permutation.)

1. Judson, Chapter 6.4 (Cosets and Lagrange's Theorem), p. 94: 17

1. Study for the midterm in recitation on Friday, March 29.
2. Read Judson, Chapter 9.2, 11.1.
3. Judson, Chapter 9.3 (Isomorphisms), p. 147: 5, 19
4. Show that if \(\phi\from G\to H\) and \(\theta\from H\to K\) are isomorphisms, then \(\phi^{-1}\) and \(\theta\circ \phi\) are isomorphisms.
5. Let \(n\in \N\). Let \(\Z_n=\{0+n\Z,1+n\Z,\dots, (n-1)+n\Z\}\) be the cyclic group of order \(n\), represented by the set of cosets of \(n\Z\). Let \(G\) be a cyclic group of order \(n\) and let \(a\) be a generator of \(G\).
   • Show that the map \(\phi\from \Z_n\to G\), \(\phi(k+n\Z)=a^k\) is well-defined.
   • Show that \(\phi\) is an isomorphism. Conclude that any cyclic group of order \(n\) is isomorphic to \(\Z_n\).
   • Show that if \(G\) and \(H\) are cyclic and \(\phi\from G\to H\) is an isomorphism, then \(\phi\) takes generators of \(G\) to generators of \(H\).
6. Let \(G\) be a group. An automorphism of \(G\) is an isomorphism from \(G\) to itself. Let \(\Aut(G)\) be the set of automorphisms of \(G\). Show that \(\Aut(G)\) is a group under composition. (We call this group the automorphism group of \(G\).)
7. • Show that \(\phi\from \Z_5\to \Z_5\), \(\phi(a)\equiv 2a\pmod{5}\) is an automorphism of \(\Z_5\).
   • List all of the automorphisms of \(\Z_5\) and show that \(\Aut(\Z_5)\) is cyclic.

This is a guide to some of the skills, definitions, examples, and theorems that we've covered in class. Try to:

1. Give an example illustrating each of the items below.
2. Find problems in the problem sets or textbook regarding each item.
3. Find theorems, propositions, and lemmas that use these items.

• Sets and functions
  • Define injective, surjective, bijective, inverse
  • Prove that a function is injective, surjective, or bijective
  • Work with set notation (e.g., \(Z(a)=\{g\in G\mid ga=ag\}\), \(\{n^2\bmod 10 \mid n\in \Z\}=\{0,1,4,5,6,9\}\))
  • Prove whether two sets are equal or unequal
• Groups and subgroups
  • Define group, subgroup, abelian, order of a group
  • Define and work with examples such as \(\Z\), \(\Z_n\), \(\Q\), \(\Q^\times\), \(\R\), \(\R^\times\), \(\C\), \(\C^\times\), \(D_n\)
  • Determine whether a given set and operation form a group
  • Determine whether a given subset of a group is a subgroup
  • Prove facts about groups using their definition
  • State the cancellation law for groups and use it to solve problems
• Cyclic groups
  • Define cyclic subgroup, cyclic group, generator, order of an element
  • Determine whether a group is cyclic
  • Determine whether an element of a group generates that group
  • Calculate the order of an element of a group
  • State the division algorithm and use it to prove facts about cyclic groups
  • Define root of unity
• Permutations
  • Define permutation, cycle, transposition, even, odd, symmetric group, alternating group
  • Multiply permutations
  • Decompose a permutation into disjoint cycles
  • Determine whether a permutation is even or odd
  • Calculate the order of a permutation
  • Prove facts about permutations
  • Calculate groups of symmetries of geometric figures like \(D_n\)
• Cosets
  • Define left coset, right coset, partition, index
  • Calculate the left cosets of a subgroup of a group
  • State Lagrange's Theorem
  • Calculate the index of a subgroup, with and without using Lagrange's Theorem
  • State the conditions equivalent to \(g_1H=g_2H\) and use them to prove facts about cosets
  • Define \(U(n)\), the multiplicative group of integers mod \(n\)
  • State Euler's Theorem and Fermat's Little Theorem
• Isomorphisms
  • Define isomorphism, isomorphic, equivalence relation, direct product, internal direct product
  • Given two groups, determine whether or not they are isomorphic
  • Determine whether a group is isomorphic to a direct product

1. Study for the midterm in recitation on Friday, March 29.
2. Read Judson, Chapter 10.1, 11.2
3. Judson, Chapter 11.3 (Homomorphisms): 2, 5
4. Let \(G\) be a finite group and let \(\phi\from G\to H\) be a homomorphism. Show that \(|\im(\phi)|\cdot |\ker(\phi)|=|G|\).

5. Use problem 4 to show that if \(G\) and \(H\) are finite groups and \(\phi\from G\to H\) is a homomorphism, then \(|\im(\phi)|\) is a common divisor of \(|G|\) and \(|H|\).

   What does this say about \(\phi\) if \(|G|\) and \(|H|\) are relatively prime?

1. Read Judson, Chapter 13.1, 16.1
2. Judson, Chapter 10 (Normal Subgroups and Factor Groups): 6, 10 (a simple group is a group that has no normal subgroups except for \(G\) and \(\{e\}\))

3. Let \(H\subset \GL_3(\R)\) be the group of matrices \[H=\left\{\begin{pmatrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix}\middle| \;x,y,z\in \R\right\}.\] This is called the Heisenberg group.

   • Check that \(H\) is a nonabelian group.
   • Show that the map \(\phi\from H\to \R^2\), \[\phi\left(\begin{pmatrix}1 & x & z \\ 0 & 1 & y \\ 0 & 0 & 1\end{pmatrix}\right)=(x,y)\] is a homomorphism from \(H\) to \(\R^2\). Calculate its kernel.
   • Use the First Isomorphism Theorem to show that the subgroup \[N=\left\{\begin{pmatrix}1 & 0 & z \\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\middle|\; z\in \R\right\}\] is normal and describe the factor group \(H/N\).

   This is an example of a group \(H\) with a normal subgroup \(N\) such that \(N\) is abelian and \(H/N\) is abelian, but \(H\) is not abelian.

4. Find an example of a group \(G\) and two subgroups \(H\) and \(K\) such that \(HK\) is not a subgroup of \(G\). (Note: By the Second Isomorphism Theorem, neither \(H\) nor \(K\) can be normal.)
5. Let \(G\) be a group and let \(N\) be a normal subgroup of \(G\). Suppose that \(G/N\) is cyclic.
   • Show that there is an element \(a\in G\) such that \(G=\langle a\rangle N\).
   • Show that there is an isomorphism \(\phi\from N \to N\) such that \(na=a\phi(n)\) for every \(n\in N\). Show that for any \(n_1, n_2\in N\), \[(a^jn_1) (a^k n_2)=a^{j+k} \phi^{k}(n_1) n_2,\] where \(\phi^k=\underbrace{\phi\circ \dots \circ \phi}_{k\text{ times}}\). (By part 1, this formula lets you compute the product of any two elements of \(G\).)
   • Show that if \(N\) is abelian and \(a\) commutes with every element of \(N\), then \(G\) is abelian.

1. Read Judson, Chapter 16.2-16.3.

2. In this problem, we will prove the following result:

   If \(G\) is a group of order 35, then \(G\) is isomorphic to \(\Z_{35}\).

   We will proceed by contradiction, so throughout the following questions, assume that \(G\) is a group of order \(35\) that is not cyclic. Most of these questions can be solved independently.

   1. Show that every element of \(G\) except the identity has order \(5\) or \(7\). Let \(T_5\) be the set of elements of order \(5\) and let \(T_7\) be the set of elements of order \(7\). (These are not subgroups!)

   2. Show that if \(H_1\) and \(H_2\) are two subgroups of order \(5\), then either \(H_1\cap H_2=\{e\}\) or \(H_1=H_2\). Use this to show that the subgroups of order \(5\) partition \(T_5\) into disjoint sets of four elements each. Show similarly that \(T_7\) can be parititioned into sets of order \(6\). Conclude that:

      \begin{align*} 4\cdot &\text{(# of subgroups of order $5$)} + \\ &6\cdot \text{(# of subgroups of order $7$)}=34 \end{align*}
   3. Use the equation above to conclude that there is at least one subgroup of order \(5\) and at least one subgroup of order \(7\).
   4. Let \(x\) be an element of order \(5\) and let \(y\) be an element of order \(7\) (i.e., \(x\in T_5\), \(y\in T_7\)). Show that \(x\) and \(y\) do not commute. (Proceed by contradiction: assume that \(xy=yx\) and show that this assumption leads to \(G\) being cyclic.)
   5. Fix a \(y\in T_7\). For each \(x\in T_5\), let \(C_x=\{x, y x y^{-1},y^2 xy^{-2}, \dots, y^6 xy^{-6}\}\). Show that these seven elements are distinct.
   6. Suppose \(w,x\in T_5\). Show that \(C_{y^kxy^{-k}}=C_x\) for any \(k\in \Z\) and that if \(C_w\cap C_x\) is not empty, then \(C_w=C_x\). Conclude that \(|T_5|\) is divisible by \(7\), and therefore that it is divisible by \(28\).
   7. Show that \(|T_7|\) is divisible by \(5\) and therefore that it is divisible by \(30\).
   8. Derive a contradiction, showing that \(G\) is cyclic.
3. Judson, Chapter 16.6 (Rings): 1abcf

1. Read Chapter 16.4, 17.1-17.2
2. Judson, Chapter 16.6 (Rings): 7 (note that \(\R\) and \(\C\) have the same cardinality, so you need to use the ring structure), 8, 18, 27
3. Let \(\Z[i]=\{a+bi\mid a,b\in \Z\}\) be the Gaussian integers and let \(I\subset \Z[i]\) be the principal ideal \(I=(1+i)\Z[i]\).
   • Recall that the norm of a complex number \(a+bi\) is defined as \(|a+bi|=\sqrt{a^2+b^2}\). What are the elements of \(I\) with norm at most \(3\)? Plot these on the complex plane.
   • What are the elements of \(\Z[i]/I\)? Write multiplication and addition tables for \(\Z[i]/I\).

1. Read Chapter 17.3. 18.1
2. Study for the quiz on Friday, May 3. The quiz will cover rings and polynomials up to what we cover in class on the 25th, i.e., the last part of Problem Set 10 and Problem Sets 11 and 12.
3. Judson, Chapter 16.6 (Rings), p. 251: 28
4. Let \(\Z[i]=\{a+bi\mid a,b\in \Z\}\) be the Gaussian integers. Let \(I=2\Z[i]\) and let \(R=\Z[i]/I\).
   • Show that \(R\) has four elements. Call these \(0\), \(1\), \(z\), and \(w\).
   • Write multiplication and addition tables for \(R\). Show that \(R\) is not an integral domain.
   • Show that \(I\) is not a prime ideal in \(\Z[i]\) by finding \(a,b\in \Z[i]\) such that \(ab\in I\) but \(a,b\not \in I\).
   • Show that \(I\) is not a maximal ideal by finding an ideal \(J\) such that \(I\subsetneq J\subsetneq \Z[i]\).
5. Judson, Chapter 17.4 (Polynomials), p. 273: 6, 7
6. Let \(I=\{p\in \R[x] \mid p(1)=0, p(2)=0\}\). Show that \(I\) is an ideal. Show that any element of \(I\) is a multiple of \(x^2-3x+2\) and thus \(I=(x^2-3x+2)\R[x]\).
7. Let \(J=\{p\in \Q[x] \mid p(\sqrt{2})=0\}\). Use the Division Algorithm and the fact that \(\sqrt{2}\) is irrational to show that any element of \(J\) is a multiple of \(x^2-2\) and thus \(J=(x^2-2)\Q[x]\).

1. Read Chapter 20, 21.1
2. Judson, Chapter 17.4 (Polynomials), p. 273: 10
3. • Show that there is an ring automorphism \(\phi\from \C\to \C\) such that \(\phi(i)=-i\) and \(\phi(r)=r\) for every \(r\in \R\).
   • Show that if \(p\in \R[x]\) is a polynomial with real coefficients and \(p(a+bi)=0\), then \(p(a-bi)=0\). That is, complex roots of real polynomials occur in conjugate pairs.
4. Let \(I=\{p\in \R[x] \mid p(\sqrt{-1})=0\}\). Show that \(I=\langle x^2+1\rangle\) and use the First Isomorphism Theorem to prove that \(\R[x]/\langle x^2+1\rangle \cong \C\).
5. Show that \(x^2+1\) is irreducible over \(\Z_3\) and conclude that \(F=\Z_3[x]/\langle x^2 + 1\rangle\) is a field. What is the order of \(F\)?
6. Let \(F=\Z_3[x]/\langle x^2 + 1\rangle\) be as above and let \(F^*=(F\setminus \{0\},\cdot)\) be the multiplicative group of \(F\). Find an element of \(F^*\) of order \(8\) and conclude that \(F^*\) is cyclic.

The final exam is scheduled for Thursday, May 16 from 10-11:50 in WWH 202. Check Albert for any last-minute schedule or location changes. You may bring one 3" by 5" index card of notes.

This is a guide to some of the skills, definitions, examples, and theorems that we've covered in class. Try to:

• Give an example illustrating each of the items below.
• Find problems in the problem sets or textbook regarding each item.
• Find theorems, propositions, and lemmas that use these items.

If you would like to ask questions before the final, please feel free to come to recitation on Friday, my office hours on Monday, or Rodion's office hours on Tuesday. I will also try to answer questions by email, but please allow up to one business day for a response.

• Sets and functions
  • Define injective, surjective, bijective, inverse
  • Prove that a function is injective, surjective, or bijective
  • Work with set notation (e.g., \(Z(a)=\{g\in G\mid ga=ag\}\), \(\{n^2\bmod 10 \mid n\in \Z\}=\{0,1,4,5,6,9\}\))
  • Prove whether two sets are equal or unequal
• Groups and subgroups
  • Define group, subgroup, abelian, order of a group
  • Define and work with examples such as \(\Z\), \(\Z_n\), \(\Q\), \(\Q^\times\), \(\R\), \(\R^\times\), \(\C\), \(\C^\times\), \(D_n\)
  • Determine whether a given set and operation form a group
  • Determine whether a given subset of a group is a subgroup
  • Prove facts about groups using their definition
  • State the cancellation law for groups and use it to solve problems
• Cyclic groups
  • Define cyclic subgroup, cyclic group, generator, order of an element
  • Determine whether a group is cyclic
  • Determine whether an element of a group generates that group
  • Calculate the order of an element of a group
  • State the division algorithm and use it to prove facts about cyclic groups
  • Define root of unity
• Permutations
  • Define permutation, cycle, transposition, even, odd, symmetric group, alternating group
  • Multiply permutations
  • Decompose a permutation into disjoint cycles
  • Determine whether a permutation is even or odd
  • Calculate the order of a permutation
  • Prove facts about permutations
  • Calculate groups of symmetries of geometric figures like \(D_n\)
• Cosets
  • Define left coset, right coset, partition, index
  • Calculate the left cosets of a subgroup of a group
  • State Lagrange's Theorem
  • Calculate the index of a subgroup, with and without using Lagrange's Theorem
  • State the conditions equivalent to \(g_1H=g_2H\) and use them to prove facts about cosets
  • Define \(U(n)\), the multiplicative group of integers mod \(n\)
  • State and use Euler's Theorem and Fermat's Little Theorem
• Isomorphisms
  • Define isomorphism, isomorphic, equivalence relation, direct product, internal direct product
  • Given two groups, determine whether or not they are isomorphic
  • Determine whether a group is isomorphic to a direct product
• Homomorphisms
  • Define homomorphism, automorphism, image, kernel
  • Use the definition to prove properties of homomorphisms
  • Calculate the image and kernel of a homomorphism
  • Find homomorphisms from one group to another
  • Determine the automorphism group of a group
• Normal subgroups and factor groups
  • Define normal subgroup, factor (or quotient) group
  • Determine whether a subgroup is normal
  • Find normal subgroups of a group
  • Prove facts about normal subgroups
  • Prove facts about factor groups
  • Calculate with factor groups.
  • Define well-defined
  • Check whether an expression is well-defined
  • State the First Isomorphism Theorem and use it to construct isomorphisms between images and quotients
  • State and use the Correspondence Theorem
• Rings
  • Define ring, ring with identity, commutative ring, integral domain, division ring, field
  • Define unit, zero divisor, characteristic
  • Decide whether a set with given operations is a ring, commutative ring, ring with identity, etc.
  • Prove facts about rings, characteristic, units, zero divisors, etc. using the definitions
• Ring homomorphisms
  • Define ring homomorphism, ring isomorphism
  • Find homomorphisms between rings
  • Determine whether two rings are isomorphic
  • Compute the kernel and image of a homomorphism
  • Prove facts about homomorphisms
• Ideals and quotients
  • Define ideal, factor (or quotient) ring
  • Decide whether a given subset of a ring is an ideal
  • Construct and calculate with quotients
  • Prove facts about ideals and quotients
  • Define maximal ideal, prime ideal, principal ideal
  • Determine whether an ideal is maximal, either by using the definition or by considering the factor ring
  • Determine whether an ideal is prime, either by using the definition or by considering the factor ring
  • State the First Isomorphism Theorem and use it to construct isomorphisms between images and quotients
• Polynomials
  • Define polynomial
  • Define degree, irreducible polynomial, root, evaluation homomorphism
  • Calculate with polynomials with coefficients in a ring. Identify units, zero divisors, etc.
  • State the Division Algorithm
  • Use the Division Algorithm to divide polynomials, find remainders, etc.
  • Use the Division Algorithm to prove facts about polynomials, polynomial rings, and ideals
  • Find a generator of an ideal in a polynomial ring
  • State and use the Factor Theorem
  • Identify irreducible polynomials in different polynomial rings
  • Describe quotients of polynomial rings and use quotients of polynomial rings to construct finite fields
  • Perform calculations in quotients of polynomial rings