Denison University Abstract Algebra on Groups PID and UF Questions

1. Let G be a set along with a binary operation ◦ such that• ◦ is associative• There exists an element e in G such that e ◦ x = x for all x ∈ G• For each a ∈ G there exists an element a0in G such that a0 ◦ a = e.Show that G is a group.

2. Prove that Aut(Q) has a single element.

3. What are all the possible disjoint cycle structures for an element in S_6? How manydistinct elements are there for each cycle structure?

Note: |S6| = 720, so you probably don’t want to write them all down.

4. Prove that the transpositions generate S_n. By this I mean that you can produce allthe elements of S_n using products of transpositions.5. Show that Z[√n] is not a PID for square-free odd n < −2.Hint 1: You might not want to do this directly... Hint 2: Once you figure out Hint 1, it might be a good idea to remind yourself howsome of your work in Z[√−5] would show this. Can you extend to odd n < −2? 6. An ideal I in a commutative ring R is finitely generated if there exists a_1, . . . a_n ∈ Rsuch that I = .Let R be a commutative ring with unity. Prove that R is a Noetherian ring if and only if every ideal inR is finitely generated.7. Let a_1, . . . a_n be elements (not all 0) of an integral domain R. A greatest commondivisor of a_1, . . . a_n is an element d of R such that:~~~d divides each of the aiin R, ~~~if c ∈ R and c divides each of the aiin R then c | d

Let R be a UFD. If a | bc and 1R is a gcd of a and b, prove that a | c.

UFDs
November 10, 2021
Circled Problems Due December 2 at 7:00am
For this sheet of problems, if you are using the fundamental theorem you can
just provide the appropriate surjective homomorphism φ. You do not need to
prove anything about φ.
√
1 Show that Z[ n] is not a PID for square-free odd n < −2. Hint 1: You might not want to do this directly... Hint 2: Once you figure√out Hint 1, it might be a good idea to remind yourself how some of your work in Z[ −5] would show this. Can you extend to odd n < −2? √ 2. Show that Z[ n] is not a PID for square-free even n < −2. √ 3. Find an element of Z[ −5] other than 6 that has two distinct factorizations into irreducibles. Prove that these elements are irreducible and that the factors in one factorization are not associates with the factors in the other factorization. √ √ 4. Let R = Z[ −5] and define the ideal Q1 = h3, 2 + −5i. Prove that Q1 is a prime ideal of R. Hint 1: You might want to start by thinking about R/h3i. Hin2: Dr. Mei might give you a better sense of what isomorphism to think about... 5 An ideal I in a commutative ring R is finitely generated if there exists a1 , . . . an ∈ R such that I = ha1 , . . . , an i. Let R be a community. Prove that R is a Noetherian ring if and only if every ideal in R is finitely generated 6. Let a1 , . . . an be elements (not all 0) of an integral domain R. A greatest common divisor of a1 , . . . an is an element d of R such that • d divides each of the ai in R • if c ∈ R and c divides each of the ai in R then c | d. √ (a) Note: in a domain, gcds are not guaranteed to exist! Show that 6 and 2 + 2 −5 √ have no greatest common divisor in Z[ −5]. (b) Let p be an irreducible element of an integral domain. Prove that 1R is a gcd of p and a if and only if p - a. c Let R be a UFD. If a | bc and 1R is a gcd of a and b, prove that a | c. 1 John & Soham Alex & Angus Erich & Lily Simon & Theodore Hardy & Reece Shuying & Will Moriah & Chase Evan & Your Choice 2 The Symmetric Group! November 16, 2021 Circled Problems Due December 2 at 7:00am 1. Prove that a nonzero ring R is not a group under multiplication. 2. Give an example of an abelian group of order 4 in which every non identity element satisfies x ◦ x = e. 3. Suppose that G is a group under operation ∗. Define a new operation # on G by a#b = b ∗ a. Prove that G is a group under #. Note: be very explicit about what you are trying to prove: what are you starting with? What do you want to conclude? How can you conclude it? 4. Let Sl2 (R) be the subset of M2 (R) that consists of matrices with determinant 1. Is Sl2 (R) a group under addition? Under multiplication? 5. (a) Write down all possible group multiplication tables for a group of order 2. (b) Write down all possible group multiplication tables for a group of order 4. Note: you do not need to prove anything, just write the tables down. 6. Show that every group G with identity e such that x ◦ x = e for all x ∈ G is abelian. 7 Let G be a set along with a binary operation ◦ such that • ◦ is associative • There exists an element e in G such that e ◦ x = x for all x ∈ G • For each a ∈ G there exists an element a0 in G such that a0 ◦ a = e. Show that G is a group. 8 Prove that Aut(Q) has a single element. 1 John & Soham Alex & Angus Erich & Lily Simon & Theodore Hardy & Reece Shuying & Will Moriah & Chase Evan & Your Choice 2 The Dihedral Group November 12, 2021 Circled Problems Due December 2 at 7:00am 1. Note that D5 , the set of symmetries of a regular pentagon, consists of 10 elements: the identity, 4 rotations, and 5 flips. Rewrite the elements of D5 as products of only two elements: a rotation ρ and a flip φ. 2. How many symmetries can you find for the unit circle? 3. Our description of the symmetries of the equilateral triangle can be elegantly rephrased using arithmetic in C. (a) The three vertices of the triangle can be thought of as numbers of the form eiak , for k = 1, 2, 3. Note that this means they live on the unit circle. Draw the triangle and the unit circle in the complex plane. Determine a1 , a2 , a3 . Hint: what length must the sides of the triangle be? (b) You can represent the rotation ρ of the triangle as a function f : C → C such that f (α) = eiθ α, for some angle θ. Determine θ, and explain your answer. (c) Find a function f : C → C that represents the flip φ. 4 Consider D3 , generated by ρ and φ. Determine φρ2 φρφρ5 in three ways: (a) First, use your triangle (I’ll believe you). (b) Second, use relationships in the multiplication table. (c) Third, rewrite this as permutation multiplication and do the multiplication. 1 John & Soham Alex & Angus Erich & Lily Simon & Theodore Hardy & Reece Shuying & Will Moriah & Chase Evan & Your Choice 2