Abstract Algebra- Basic problems for MT2

Instructions:

• Provide clear and complete reasoning for every problem.
• Use results from lectures and notes provided, referencing definitions where appropriate.
• The problems are divided into six parts, each focusing on specific topics covered in the notes.
• Mathematical expressions are enclosed within $...$ for inline math and $$...$$ for display math.
• Total of 30 problems are provided, each with meaningful content and appropriate difficulty.

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Part I: Group Actions and Permutations

Problem 1: Orbits and Stabilizers in Symmetric Groups

Let, the symmetric group on 4 elements, act on the setby permutation.

1. (a) For the element, determine the orbitunder the action of.

2. (b) Compute the stabilizer subgroup.

3. (c) Calculate the sizes ofand, and verify the Orbit-Stabilizer Theorem.

Hint: Use the definitions of orbit and stabilizer, and recall that.

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Problem 2: Group Action on Subsets

Letact on the setof all 2-element subsets ofby permutation.

1. (a) How many elements are in?

2. (b) For the subset, determine its orbit under the action of.

3. (c) Find the stabilizer subgroupand compute its order.

4. (d) Verify the Orbit-Stabilizer Theorem for this action.

Hint: Consider how permutations affect subsets and calculate accordingly.

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Problem 3: Orbits in Group Actions

Letbe the subgroup ofconsisting of all permutations that fix the element(i.e.,).

1. (a) Describe all elements of.

2. (b) Determine the orbits ofacting on.

3. (c) Find the orbit ofand compute the size of.

Hint: Use the fact that permutations infixand permute the other elements.

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Problem 4: Group Actions and Equivalence Relations

Let a groupact on a set. Define a relationonbyif there existssuch that.

1. (a) Prove thatis an equivalence relation.

2. (b) Show that the equivalence classes underare precisely the orbits ofon.

Hint: Verify the properties of reflexivity, symmetry, and transitivity using the group action.

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Problem 5: Orbit-Stabilizer Theorem Application

Letact onby permutation:.

1. (a) Determine the size of.

2. (b) For the element, find.

3. (c) Computeand verify the Orbit-Stabilizer Theorem.

Hint: Consider the action ofon ordered pairs.

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Part II: Definitions and Equivalence of Group Actions

Problem 6: Group Actions via Homomorphisms

1. (a) Define what it means for a groupto act on a setvia a function.

2. (b) Show that this action defines, for each, a bijection.

3. (c) Prove that the mapdefined byis a group homomorphism.

Hint: Use the properties of the group action and the composition of functions.

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Problem 7: Equivalence of Definitions

1. (a) Show that any group homomorphismdefines a group action ofonvia.

2. (b) Prove that the two definitions of group action (viaand via) are equivalent.

Hint: Construct the correspondence betweenandexplicitly.

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Problem 8: Kernels of Group Actions

Letact on a set, and letbe the associated homomorphism.

1. (a) Define the kernel of the action,.

2. (b) Prove that.

3. (c) Show thatis a normal subgroup of.

Hint: Use the properties of group homomorphisms.

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Problem 9: Faithful Group Actions

1. (a) Define what it means for a group action to be faithful.

2. (b) Prove that the action is faithful if and only if, whereis the identity in.

3. (c) Give an example of a faithful and a non-faithful group action.

Hint: Consider the effect of elements inon.

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Problem 10: Left Regular Action Example

Letact on itself via the left regular action.

1. (a) Describe explicitly how the left regular action is defined for.

2. (b) Write down the permutation representation ofcorresponding to this action.

3. (c) Show that this action is faithful.

Hint: The left regular action is given by.

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Part III: Left Regular Action and Cayley’s Theorem

Problem 11: Cayley’s Theorem Proof

1. (a) State Cayley’s Theorem.

2. (b) Provide a detailed proof of Cayley’s Theorem using the left regular action.

Hint: Show that the mapping fromtovia the left regular action is an injective homomorphism.

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Problem 12: Left Regular Action of a Non-Abelian Group

Letbe the symmetric group.

1. (a) Describe the left regular action ofon itself.

2. (b) Write down the permutation matrices corresponding to the action.

3. (c) Show thatis isomorphic to a subgroup of.

Hint: Since, the left regular action mapsinto.

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Problem 13: Kernel of Left Regular Action

Letbe any group acting on itself by left multiplication.

1. (a) Determine the kernel of this action.

2. (b) Conclude whether the action is faithful.

Hint: Consider whether any non-identity element fixes all elements under left multiplication.

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Problem 14: Action ofon Itself

Let, and define an action ofon itself by.

1. (a) Is this action faithful?

2. (b) Find the associated homomorphism.

3. (c) Determine the kernel of.

Hint: Analyze whether different elements ofinduce different permutations.

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Problem 15: Cayley’s Theorem Application

Let, the dihedral group of order 8.

1. (a) Describe the elements of.

2. (b) Use the left regular action to embedinto.

3. (c) Show that this embedding is injective.

Hint: Consider the permutations ofinduced by left multiplication.

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Part IV: Burnside’s Lemma and Counting Colorings

Problem 16: Burnside’s Lemma Statement and Proof

1. (a) State Burnside’s Lemma.

2. (b) Provide a proof of Burnside’s Lemma using double counting.

Hint: Count the number of pairssuch that.

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Problem 17: Coloring Vertices of a Square

Consider coloring the vertices of a square usingcolors, where two colorings are considered the same if one can be obtained from the other by a rotation or reflection (the action of).

1. (a) Determine the groupacting on the colorings.

2. (b) Use Burnside’s Lemma to calculate the number of distinct colorings.

3. (c) Compute this number explicitly for.

Hint: For each element, find the number of colorings fixed by.

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Problem 18: Coloring Edges of a Cube

Consider coloring the edges of a cube usingcolors, up to rotational symmetries.

1. (a) Describe the rotation groupacting on the cube.

2. (b) Determine the number of elements in.

3. (c) Use Burnside’s Lemma to find the number of distinct colorings when.

Hint: Identify the types of rotations and calculate fixed colorings.

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Problem 19: Coloring with Cyclic Group Actions

Consider a necklace withbeads, andacting by rotation.

1. (a) Describe the action ofon the set of colorings.

2. (b) Use Burnside’s Lemma to compute the number of distinct colorings withcolors.

3. (c) Calculate this number explicitly forand.

Hint: For each rotation, determine the number of colorings it fixes.

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Problem 20: Fixed Points under Group Action

Letact on a finite set, and suppose that for every, the number of fixed pointsis known.

1. (a) Use Burnside’s Lemma to compute the number of orbits ofon.

2. (b) Ifandfor all, and, find the number of orbits.

Hint: Apply the formula from Burnside’s Lemma.

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Part V: Rings, Fields, Units, and Zero-Divisors

Problem 21: Units and Zero-Divisors in Rings

1. (a) Define a unit and a zero-divisor in a ring.

2. (b) Prove that in any ring, the set of units and the set of zero-divisors (excluding zero) are disjoint.

3. (c) Provide an example of a ring where elements are neither units nor zero-divisors.

Hint: Consider the ringand its elements.

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Problem 22: Units in Finite Rings

Let.

1. (a) List all units in.

2. (b) List all zero-divisors in.

3. (c) Verify thatconsists of units, zero-divisors, and zero.

Hint: An elementis a unit if.

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Problem 23: Units in Gaussian Integers

Consider the ringof Gaussian integers.

1. (a) Define what it means for an element to be a unit in.

2. (b) Find all units in.

3. (c) Prove thatis not a unit in.

Hint: Use the norm.

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Problem 24: Zero-Divisors in Rings

1. (a) Give an example of an infinite ring with zero-divisors.

2. (b) Show that in the ring, there exist zero-divisors.

3. (c) Provide an explicit example of two non-zero matricesandsuch that.

Hint: Consider matrices where the product results in the zero matrix.

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Problem 25: Division Algorithm in

1. (a) State the Division Algorithm in.

2. (b) Given integersand, find integersandsuch thatand.

3. (c) Verify that the values ofandsatisfy the conditions.

Hint: Perform integer division and find the remainder.

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Problem 26: Division Algorithm in

1. (a) State the Division Algorithm for polynomials in.

2. (b) Dividebyand find the quotient and remainder.

3. (c) Verify that the degree of the remainder is less than the degree of.

Hint: Be careful with integer coefficients during polynomial division.

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Problem 27: Division Algorithm in

Letandin.

1. (a) Use the Division Algorithm into findandsuch thatwith.

2. (b) Compute the norm ofand verify the inequality.

3. (c) Show all steps of your calculation.

Hint: Find the closest Gaussian integer to.

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Problem 28: Units and Zero-Divisors in

Letbe a positive integer.

1. (a) Describe the units in the ring.

2. (b) Explain whyis a zero-divisor inifand.

3. (c) For, list all units and zero-divisors in.

Hint: Use properties of modular arithmetic.

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Problem 29: Units in Polynomial Rings over Fields

Let, the ring of polynomials with rational coefficients.

1. (a) Determine all units in.

2. (b) Is the polynomiala unit in? Justify your answer.

3. (c) Explain why polynomials of degree zero are units if they are non-zero.

Hint: Units inare the invertible elements under multiplication.

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Problem 30: Non-Commutative Rings and Units

Consider the ring.

1. (a) Determine the units in.

2. (b) Explain whyis non-commutative.

3. (c) Is every non-zero element ofeither a unit or a zero-divisor? Justify your answer.

Hint: Recall that invertible matrices have non-zero determinants.

Midterm 2 Practice Problems: Division Algorithm/GCD of Polynomials and Burnside’s Lemma

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Instructions:

• Provide clear and complete reasoning for every problem.
• Use results from lectures and the provided notes, referencing definitions where appropriate.
• The problems are designed to be challenging and integrate multiple concepts from your course.
• Mathematical expressions are enclosed within $...$ for inline math and $$...$$ for display math.
• Each section contains 10 problems focused on the specified topic.
• Problems are intended to be comprehensive and computationally intensive, suitable for exam preparation.

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Part I: Division Algorithm and GCD of Polynomials

Problem 1: Division Algorithm in

Letandin.

1. (a) Use the Division Algorithm to divideby, and find the quotientand remainder.

2. (b) Verify that.

3. (c) Determine the degrees ofand.

Hint: Be careful with polynomial long division involving rational coefficients.

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Problem 2: Euclidean Algorithm in

Letandin.

1. (a) Use the Euclidean Algorithm to compute the greatest common divisor.

2. (b) Expressas a linear combination ofand.

3. (c) Determine whetherandare coprime.

Hint: Perform successive divisions and track the remainders.

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Problem 3: Factorization in

Letin.

1. (a) Factorcompletely over.

2. (b) Use the Division Algorithm to confirm one of the factors.

3. (c) Find the.

Hint: Consider factoringinto quadratic or linear factors.

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Problem 4: Irreducibility and GCD

Letandin.

1. (a) Show thatis irreducible over.

2. (b) Compute the.

3. (c) Explain whyandare coprime.

Hint: Use Eisenstein’s Criterion for irreducibility.

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Problem 5: Division Algorithm inwith Modulo Arithmetic

Letandin.

1. (a) Perform the division ofbyin.

2. (b) Find the quotient and remainder.

3. (c) Verify the Division Algorithm in this ring.

Hint: Remember that coefficients are modulo.

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Problem 6: GCD of Polynomials over Finite Fields

Letandin, whereis the finite field with two elements.

1. (a) Computein.

2. (b) Factorandcompletely over.

3. (c) Determine whetherandare coprime.

Hint: In, addition and subtraction are the same.

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Problem 7: Application of Remainder Theorem

Letin.

1. (a) Use the Remainder Theorem to find the remainder whenis divided by.

2. (b) Confirm your result using the Division Algorithm.

3. (c) Determine ifis a factor of.

Hint: Evaluatefor the remainder.

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Problem 8: Extended Euclidean Algorithm

Letandin.

1. (a) Use the Extended Euclidean Algorithm to find polynomialsandsuch that.

2. (b) Interpret the result in terms of.

3. (c) Determine ifandare coprime.

Hint: The Extended Euclidean Algorithm generalizes to polynomials.

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Problem 9: Content of a Polynomial

Define the content of a polynomialas the greatest common divisor of its coefficients.

Let.

1. (a) Compute the content of.

2. (b) Find the primitive part of(divide by its content).

3. (c) Factor the primitive part over.

Hint: Factor out theof the coefficients first.

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Problem 10: Gauss’s Lemma and GCD

Letandin.

1. (a) Show that the productis primitive.

2. (b) Compute.

3. (c) Use Gauss’s Lemma to explain your findings.

Hint: Recall that the product of primitive polynomials is primitive.

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Part II: Burnside’s Lemma Applications (Without Polyhedra)

Problem 11: Coloring a Necklace with Beads

Consider a necklace withbeads, andcolors available to color each bead. Two necklaces are considered the same if one can be obtained from the other by rotation (no reflections).

1. (a) Describe the cyclic groupacting on the set of necklaces.

2. (b) Use Burnside’s Lemma to compute the number of distinct necklaces.

3. (c) Calculate this number explicitly forand.

4. (d) Determine the number of necklaces fixed by a rotation of.

Hint: For each rotation, compute the number of colorings it fixes.

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Problem 12: Coloring Squares on a Grid

Consider agrid of squares, and you havecolors to color each square. Two colorings are considered the same if they can be obtained from one another by rotation of(only considering rotation by).

1. (a) Describe the groupacting on the set of colorings.

2. (b) Use Burnside’s Lemma to find the number of distinct colorings.

3. (c) Calculate this number explicitly for.

4. (d) Determine the number of colorings fixed by therotation.

Hint: The grouphas two elements: the identity and therotation.

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Problem 13: Permutations and Colorings

Letact on the set of colorings of four objects usingcolors, where two colorings are considered the same if they can be obtained from one another by a permutation in.

1. (a) Describe the action ofon the colorings.

2. (b) Use Burnside’s Lemma to compute the number of distinct colorings.

3. (c) Compute this number explicitly for.

4. (d) Determine the number of colorings fixed by a permutation of cycle type.

Hint: Identify the cycle types and compute fixed colorings accordingly.

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Problem 14: Coloring Vertices of a Polygon

Consider a regular-gon in the plane, and you want to color its vertices usingcolors. Two colorings are considered the same if they can be obtained from one another by rotation (no reflections).

1. (a) Describe the cyclic groupacting on the vertices.

2. (b) Use Burnside’s Lemma to compute the number of distinct vertex colorings.

3. (c) Calculate this number explicitly forand.

4. (d) Determine the number of colorings fixed by a rotation of.

Hint: The rotation group haselements corresponding to rotations.

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Problem 15: Coloring Strings with Repetition

Consider strings of lengthformed from an alphabet ofletters. Two strings are considered the same if one can be transformed into the other by reversing the string (palindrome consideration).

1. (a) Describe the groupacting on the set of strings.

2. (b) Use Burnside’s Lemma to find the number of distinct strings under this equivalence.

3. (c) Compute this number explicitly forand.

4. (d) Determine the number of strings fixed under the reversal.

Hint: The group has two elements: the identity and the reversal.

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Problem 16: Coloring Cells in a Circular Array

Consider a circular array ofcells, each of which can be colored withcolors. Two colorings are considered the same if they can be rotated or reflected (dihedral group).

1. (a) Describe the dihedral groupacting on the colorings.

2. (b) Use Burnside’s Lemma to compute the number of distinct colorings.

3. (c) Calculate this number explicitly forand.

4. (d) Determine the number of colorings fixed by a reflection.

Hint: Account for both rotations and reflections in your calculations.

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Problem 17: Coloring Positions in a Dance Circle

In a dance circle withpositions, each dancer wears a dress of one ofcolors. Two arrangements are considered the same if they can be obtained from one another by rotation.

1. (a) Describe the cyclic groupacting on the arrangements.

2. (b) Use Burnside’s Lemma to compute the number of distinct arrangements.

3. (c) Calculate this number explicitly forand.

4. (d) Determine the number of arrangements fixed by a rotation of.

Hint: Similar to necklace counting but applied to dancers.

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Problem 18: Coloring Squares in a Magic Square

Consider arranging numbers fromtoin angrid such that two arrangements are considered the same if they can be obtained from one another by rotation or reflection.

1. (a) Describe the groupacting on the set of arrangements.

2. (b) Use Burnside’s Lemma to estimate the number of distinct arrangements (this may be complex; focus on understanding the method).

3. (c) Discuss the difficulties in computing this number explicitly.

4. (d) For a simplified case with, compute the number of distinct arrangements.

Hint: This problem is more theoretical; consider permutations and symmetries.

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Problem 19: Coloring Cells in a Honeycomb

Consider a finite honeycomb lattice in the plane withhexagonal cells, each of which can be colored usingcolors. Two colorings are considered the same if they can be obtained from one another by translations and rotations within the plane.

1. (a) Describe the symmetry groupacting on the colorings.

2. (b) Use Burnside’s Lemma to compute the number of distinct colorings.

3. (c) Explain why this calculation may be complex.

4. (d) For a simplified case withcells forming a triangle, compute the number of distinct colorings for.

Hint: Focus on the symmetries of the small configuration.

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Problem 20: Coloring Lattice Points in the Plane

Consider a finite grid ofpoints in the plane, and each point can be colored usingcolors. Two colorings are considered the same if they can be obtained from one another by a rotation ofabout the center of the grid.

1. (a) Describe the groupacting on the colorings.

2. (b) Use Burnside’s Lemma to compute the number of distinct colorings.

3. (c) Compute this number explicitly forand.

4. (d) Determine the number of colorings fixed by therotation.

Hint: Only consider the identity and therotation.

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End of Practice Problems