Exploring Abstract Algebra with Mathematica

Prerequisites and Goals in Exploring Abstract Algebra with Mathematica

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This page gives the Prerequisites and Goals sections for each lab in Exploring Abstract Algebra with Mathematica. Either scroll through the material or use a link below to jump to the desired portion. Alternatively, this material is also available as a Mathematica notebook.

• Group Lab 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14
• Ring Lab 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13

Part I -- Group Labs

• Lab 0. Getting started with Mathematica

  • 0.1 Prerequisites - There are no prerequisites for this lab.
  • 0.2 Goals for this lab - This lab is intended to introduce a few of the basics of Mathematica, as well as introduce a few rudimentary algebraic ideas.

• Group Lab 1. Using symmetry to uncover a group

  • 1.1 Prerequisites - There are no prerequisites for this lab.
  • 1.2 Goals for this lab - In this lab, we will try to discover some of the basic properties of a group by considering the symmetries of a regular triangle.

• Group Lab 2. Determining the symmetry group of a given figure

  • 2.1 Prerequisites - Though not absolutely necessary, it would be useful if you completed Group Lab 1 before attempting this lab.
  • 2.2 Goals for this lab - In this lab, we will continue to look at symmetries. This will resume where Lab One left off. The goal is to find the complete list of symmetries (via permutations) for a variety of (more or less) random figures. When "complete," this list should comprise the "symmetry group" of the object.

• Group Lab 3. Is this a group?

  • 3.1 Prerequisites - To complete this lab, you should have already seen the definition of a group and be familiar with the basic group properties: being closed, identity, inverse, and associativity (and commutativity).
  • 3.2 Goals for this lab - Given a set of elements, and some operation on the elements, will this form a group? This lab will explore a number of pairs consisting of a set and an operation. A Cayley table for the set and operation is presented and the user is asked which of the defining properties of a group hold for this pair.

• Group Lab 4. Let's get these orders straight

  • 4.1 Prerequisites - To complete this lab you should be familiar with the basic definition of a group. You should also have seen the definition of the order of an element in a group. (Recall that the order of an element g of a finite group G is the least positive integer k such that is equal to the identity of G.)
  • 4.2 Goals for this lab - In this lab, we will look at issues regarding the order of groups and their elements. First we consider the relationship between the order of g and the order of the inverse of g. We then look at the distribution of the orders of elements in , followed by an inspection of which elements share a common order. We then begin an exploration regarding the probability that an arbitrary element of will generate the whole group. Finally, we consider the order of the group (the multiplicative units of ) and try to find an expression for this order in terms of n.

• Group Lab 5. Subversively grouping our elements

  • 5.1 Prerequisites - To complete this lab, you should be familiar with the definition of a subgroup of a group.
  • 5.2 Goals for this lab - What constitutes a subgroup? What elements are necessary before a set can be considered a subgroup? What do the subgroups of look like? What about the subgroups of ? What is the probability that a randomly chosen subset of elements from will actually be a subgroup? What elements of will guarantee closure to the full group? These are some of the questions that will be explored in this lab.

• Group Lab 6. Cycling through the groups

  • 6.1 Prerequisites - Other than the basic definitions related to a group, there are no prerequisites.
  • 6.2 Goals for this lab - We first look at what it means for a group to be cyclic and try to determine the generators when it is. We will try to classify the cyclicity of the groups , , and , and determine the set of generators when they are cyclic. Next, we consider the case when the direct product of and yields a cyclic group. Finally, we look at some of the cyclic subgroups of the infinite additive group of integers, Z.

• Group Lab 7. Permutations

  • 7.1 Prerequisites - To complete this lab, you should have a good understanding of functions, including "right to left" composition. It is not necessary to have completed any previous labs to attempt this one.
  • 7.2 Goals for this lab - We will look at the notion of a permutation and how a group can be formed with permutations. Additionally, we will look at properties of permutations and consider different ways of rewriting a permutation to gain insights regarding products and orders.

• Group Lab 8. Isomorphisms

  • 8.1 Prerequisites - To complete this lab, you should be familiar with the basic properties of groups to be able to compare the various pairs of groups that you will be asked to examine. No previous labs are necessary.
  • 8.2 Goals for this lab - This lab explores the notion of isomorphisms. First we define an isomorphism and then we see how one can be constructed. Next, we explore when two groups are isomorphic.

• Group Lab 9. Automorphisms

  • 9.1 Prerequisites - To complete this lab, you should have completed Group Lab 8.
  • 9.2 Goals for this lab - This lab continues the exploration of isomorphisms begun in the previous lab. In this lab, we look at a collection of isomorphisms from a group to itself and ask what kind of structure, if any, might be present.

• Group Lab 10. Direct products

  • 10.1 Prerequisites - To complete the last section of this lab, you should have completed the lab on isomorphisms (Group Lab 8); otherwise, there are no prerequisites.
  • 10.2 Goals for this lab - This lab explores the direct product of two groups. First we define the concept of a direct product and how to determine its order. Next we determine the order of an element in a direct product. We also consider when a direct product might be cyclic, given its factors are cyclic. Finally, we consider when U groups are isomorphic to direct products of other U groups.

• Group Lab 11. Cosets

  • 11.1 Prerequisites - This lab is self-contained. No prior labs need to be completed to attempt this one. One should be familiar with the basic groups such as , and .
  • 11.2 Goals for this lab - This lab explores the notion of cosets. We will look at how cosets are determined, the different types of cosets and some of the properties of cosets.

• Group Lab 12. Normality and factor groups

  • 12.1 Prerequisites - Before attempting this lab you should complete Group Lab 11 (on cosets).
  • 12.2 Goals for this lab - After defining a normal subgroup, we try to make some sense of finding an operation to act on cosets. This leads to the development of the quotient or factor group. We conclude by illustrating why normality is important in constructing the factor group.

• Group Lab 13. Group homomorphisms

  • 13.1 Prerequisites - It is presumed that the reader is familiar with isomorphisms (Group Lab 8) and cosets (Group Lab 11).
  • 13.2 Goals for this lab - This lab explores the concept of group homomorphisms. The ultimate goal is for the reader to understand the relationship between the domain, kernel and image of a homomorphism through the Fundamental Theorem of Group Homomorphisms.

• Group Lab 14. Rotational groups of regular polyhedra

  • 14.1 Prerequisites - To complete this lab, you should know how a group can be generated from a set of elements and a binary operation. You should also be familiar with Euler angles (see the Rotations Lab on the CD for a review) and group actions.
  • 14.2 Goals for this lab - The goal of this lab is to show how to generate the rotational groups of polyhedra.

Part II -- Ring Labs

• Ring Lab 1. An introduction to rings and Ringoids

  • 1.1 Prerequisites - There are no prerequisites for this lab, although a brief introduction to the terminology related to rings may be beneficial.
  • 1.2 Goals for this lab - This lab is intended to introduce the user both to the mathematical concept of the structure of a ring as well as the corresponding Mathematica structure Ringoid. Various properties that a ring must and can have are introduced and the Mathematica commands to explore these properties are illustrated.

• Ring Lab 2. An introduction to rings: part two

  • 2.1 Prerequisites - You should complete Ring Lab 1 before attempting this lab.
  • 2.2 Goals for this lab - The goals of this lab are to familiarize you with several important types of rings and to make you aware of how to work with these rings in the context of the AbstractAlgebra packages.

• Ring Lab 3. An ideal part of rings

  • 3.1 Prerequisites - Before attempting this lab, you should have completed Ring Lab 1. You should also be familiar with cosets of normal subgroups.
  • 3.2 Goals for this lab - The goal of this lab is to develop the concept of an ideal through examples, leading one to discover some of the properties of ideals. Quotient rings are also introduced through the examples.

• Ring Lab 4. What does look like?

  • 4.1 Prerequisites - Prior to working on this lab, you should be familiar with the term ideal through discussions in class or from Ring Lab 3: An ideal part of rings. One should also be familiar with an integral domain, field, and the characteristic of a ring.
  • 4.2 Goals for this lab - The goal of this lab is to explore the quotient structure of the Gaussian integers modulo an ideal generated by an arbitrary Gaussian integer.

• Ring Lab 5. Ring homomorphisms

  • 5.1 Prerequisites - Before you start this lab, you should be familiar with Ringoids and the ideas found in Ring Labs 1 and 2, as well as normal subgroups and ideals.
  • 5.2 Goals for this lab - This lab explores the notion of a ring homomorphism. First we define one, and then we see how one can be constructed.
    
    This lab is designed to be independent of the group labs on isomorphisms and homomorphisms. If you have done them, you can skip the first section of this lab, except to evaluate the inputs that define Morphoids f, g, and w.

• Ring Lab 6. Polynomial rings

  • 6.1 Prerequisites - To work on this lab, you need only a cursory familiarity with Ringoids, mostly just .
  • 6.2 Goals for this lab - The goal of this lab is to help one discover some of the basic properties of polynomial algebra over a ring, through division and the GCD function. Factorization is discussed in detail in Ring Lab 7.

• Ring Lab 7. Factoring and irreducibility

  • 7.1 Prerequisites - Before working on this lab, you should be familiar with polynomial arithmetic over integral domains. No previous labs need to be completed prior to attempting this lab.
  • 7.2 Goals for this lab - The goal of this lab is to introduce some of the tools available for polynomial factorization over a variety of rings.

• Ring Lab 8. Roots of unity

  • 8.1 Prerequisites - No other lab needs to be completed before attempting this lab. However, experience with cyclic groups (see Group Lab 6) may prove beneficial.
  • 8.2 Goals for this lab - The main goal of this lab is to become familiar with the roots of unity, the roots of polynomials of the form .

• Ring Lab 9. Cyclotomic polynomials

  • 9.1 Prerequisites - Before working on this lab, you should have completed Ring Lab 8, on roots of unity.
  • 9.2 Goals for this lab - The goal for this lab is to formulate recursive and non-recursive definitions of the cyclotomic polynomials and discover some of the properties of these polynomials.

• Ring Lab 10. Quotient rings of polynomials

  • 10.1 Prerequisites - To complete this lab, you should be familiar with the ring of polynomials over a field, the division property for polynomials over a field, and the definitions of homomorphism, kernel, and ideal. Finally, you should be familiar with the First Isomorphism Theorem for ring homomorphisms (Ring Lab 5).
  • 10.2 Goals for this lab - Extensions of finite fields are generally motivated by the need to solve polynomial equations. These extensions are actually quite concrete in the sense that they arise from quotient rings, where the ideal from which cosets are formed is the kernel of a familiar homomorphism. In this case, the homomorphism is the remainder function for division by a fixed polynomial (the modulus). In this lab we will introduce quotient rings, and in Ring Lab 11 we will explore how they are used to construct roots of polynomials.

• Ring Lab 11. Quadratic field extensions

  • 11.1 Prerequisites - To complete this lab, you should be familiar with the construction of quotient rings of the ring of polynomials over a field F. You should also be familiar with irreducible polynomials over a field. This lab does not presume any other prior knowledge of field extensions. Doing Ring Lab 10 first would be helpful, but not necessary.
  • 11.2 Goals for this lab - The goal of this lab is to provide some experience in working with quadratic field extensions in order to make the general study of finite field extensions easier to understand.

• Ring Lab 12. Factoring in

  • 12.1 Prerequisites - One should have an elementary understanding of divisors and factoring with integers. It may also be helpful to be familiar with the ring .
  • 12.2 Goals for this lab - The goal of this lab is to explore the notion of factoring numbers in for various integers d. In particular, we want to see when this factorization is unique (in some sense) and when it is not.

• Ring Lab 13. Finite fields

  • 13.1 Prerequisites - To complete this lab, you should be familiar with the construction of a quotient ring of a ring of polynomials over a field F, as described in Ring Lab 10. You should also be familiar with Morphoids of rings.
  • 13.2 Goals for this lab - The goal of this lab is to introduce some of the properties of finite fields. Until now, we have mostly explored the general situation of polynomials over a ringoid mod an arbitrary polynomial. Now we assume that the base ringoid is a field and the polynomial is irreducible over that field. These assumptions are made easier to enforce by using the FiniteFields package in AbstractAlgebra.

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Prepared by Al Hibbard. Most recent update: 6/16/2006. This page has been viewed 306 times since June 16, 2006. The entire EAAM web site has had 669,318 hits since July 23, 2002.