Descriptions of the labs in Exploring Abstract Algebra with Mathematica

This section gives brief descriptions of the labs available in Exploring Abstract Algebra with Mathematica. The group labs are given first, followed by the ring labs.

Group Labs in EAAM

Group Lab 1. Using symmetry to uncover a group -- This lab explores the underlying definitions of a group by looking at the symmetries of an equilateral triangle.
Group Lab 2. Determining the symmetry group of a given figure -- The focus of this lab is to determine the symmetry group of a figure chosen randomly from a list of regular polygons and "cyclic" objects.
Group Lab 3. Is this a group? -- This lab randomly presents a Cayley table of one of 20 "possible groups." The goal is to determine which of the defining properties of a group are reflected in the Cayley table to see if it represents a group.
Group Lab 4. Let's get these orders straight! -- This lab looks at the order of an element and its inverse, the distribution of the orders of the elements in Z_n, investigates the probability that an element in Z_n has order n and also explores the group U_n (the units in Z_n).
Group Lab 5. Subversively grouping our elements -- This lab explores the notion of a subgroup, including looking at the subgroups of Z_n and U_n, calculating the probability that a random subset of Z_n is a subgroup and determining what elements in a subset are necessary so that the closure yields the whole group.
Group Lab 6. Cycling through the groups -- Here we focus on the notion of a cyclic group and its subgroup structure. We also look at the determining when the direct sum of Z_m and Z_n yields a cyclic group.
Group Lab 7. Permutations -- This lab looks at the definition of a permutation, how to perform computations and explore properties. We also look at some applications of permutations.
Group Lab 8. Isomorphisms -- Here we look at the definition of an isomorphism and then use various visual mechanisms to try to determine when two groups are or are not isomorphic.
Group Lab 9. Automorphisms -- In this lab, we look at the group of automorphisms of Z_n and also look at inner automorphisms.
Group Lab 10. Direct Products -- The notion of direct products (sums) are introduced and we determine the order of elements in a direct product. We also try to determine when the direct product of cyclic groups is still cyclic. We also look for isomorphisms between some U_n groups.
Group Lab 11. Cosets -- This lab explores the definition and properties of cosets.
Group Lab 12. Normality and Factor groups -- A normal group is defined and explored and then used to define and explore factor groups.
Group Lab 13. Homomorphisms -- This lab explores group homomorphisms.
Group Lab 14: Rotational groups of regular polyhedra -- Here we look at how to generate the rotational groups of several polyhedra.

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Ring Labs in EAAM

Ring Lab 1. An Introduction to Ringoids and Rings -- This introduces some of the definitions and properties of rings.
Ring Lab 2. An Introduction to Rings: part two -- Guess what this is about!
Ring Lab 3. An ideal part of rings -- This explores the notion of an ideal and properties related to it.
Ring Lab 4. What does Z[i]/<a+bi> look like? -- This lab focuses on the Gaussian integers mod an ideal generated by some Gaussian integer.
Ring Lab 5. Ring homomorphisms -- This lab looks at ring homomorphisms, the First Isomorphism Theorem, and the Chinese Remainder Theorem.
Ring Lab 6. Polynomial rings -- Some basic properties of polynomial rings are introduced and explored.
Ring Lab 7. Factoring and irreducibility -- What does it mean to factor a polynomial? Various definitions and techniques are introduced.
Ring Lab 8. Roots of unity -- This lab focuses on the polynomial x^n - 1 and explores graphically the zeros of this polynomial, in particular seeing how the zeros are related to the factors and how the group U_n springs out of this.
Ring Lab 9. Cyclotomic polynomials -- This lab focuses on cyclotomic polynomials and the many properties related to them.
Ring Lab 10. Quotient rings of polynomials -- The notion of a quotient ring over a polynomial is introduced in this lab.
Ring Lab 11. Quadratic field extensions -- This lab continues the last by looking more closely at quotient rings modulo a quadratic polynomial where the result is a field.
Ring Lab 12. Factoring in Z[Sqrt[d]] -- This lab focuses on the rings Z[Sqrt[d]] and pursues the notion of divisibility and factoring in such rings. Several rings are illustrated as failing being a UFD.
Ring Lab 13. Finite Fields -- This lab continues the ideas formulated in lab 11 by looking at Galois fields and properties related to them.

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Prepared by Al Hibbard. Most recent update: 6/16/2006. This page has been viewed 290 times since June 16, 2006. The entire EAAM web site has had 669,481 hits since July 23, 2002.