Exploring Abstract Algebra with Mathematica
Al Hibbard - Central College -
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Ken Levasseur - UMass-Lowell -
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Labs for Group Theory: Exploring Abstract Algebra with Mathematica
What is given below is an outline of a presentation given by Al Hibbard at the "
Innovations in Teaching Abstract Algebra" session of the Joint Mathematics
Meetings, held at San Diego, January, 1997.
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Lab 1. Using symmetry to uncover a group
In this lab, we will try to discover the properties of a group by considering
a regular n-gon.
Lab 2. Determining the symmetry group of a given figure
In this lab, we will continue to look at symmetries. This will resume where
Lab One left off. The goal will be 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.
Lab 3. Is This a Group?
Given a set of elements and some operation on the elements, will this form a
group? This lab will explore a number of pairs of sets with an operation and
determine which properties of a group hold. A Cayley table is presented and
the user is asked which of the defining properties of a group hold.
Lab 4. Let's get these orders straight!
In this lab, we will look at issues regarding the order of elements and
groups. 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 Z[n], followed by an inspection of which elements share a common
order. We then begin an exploration regarding the probability that an
arbitrary element of Z[n] will generate the whole group. Finally, we consider
the order of the group U[n] (the multiplicative units of Z[n]) and try to find
an expression for this order in terms of n.
Lab 5. Subversively grouping our elements
What constitutes a subgroup? What elements are necessary? What do the
subgroups of Z[n] look like? What about U(n)? What is the probability of a
random subset of elements from Z[n] actually being a subgroup? What elements
of Z[n] will guarantee a closure of the full group? These are some of the
questions that will be explored in this lab.
Lab 6. Cycling through the groups
What constitutes a subgroup? What elements are necessary? What do the
subgroups of Z[n] look like? What about U(n)? What is the probability of a
random subset of elements from Z[n] actually being a subgroup? What elements
of Z[n] will guarantee a closure of the full group? These are some of the
questions that will be explored in this lab.
Lab 7. Permutations
Here we will look at the notion of a permutation, viewing one both
geometrically and symbolically. Additionally, we look at properties of
permutations and consider different ways of rewriting a permutation to gain
insights regarding products and orders.
Lab 8. Isomorphisms
This lab explores the notion of isomorphisms. First we define one and then see
how one can be constructed. Next we explore when two groups are isomorphic.
Finally we look at a collection of isomorphisms from a group to itself and ask
what kind of structure, if any, might be present.
Lab 9. Direct Products
This lab explores the direct product of two groups. First we define the
concept and determine is 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.
Lab 10. Cosets
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.
Lab 11. Normality and Factor groups
This lab explores the notion of normality of subgroups and how this is
necessary in the construction of factor groups.
Lab 12. Homomorphisms
In this lab, the definition and properties of homomorphisms are explored..
Other illustrations
Here, we just illustrate some other functionality of the underlying package.
This is by no means exhaustive.
Ring stuff
Since the focus of the talk was on the labs for group theory, we do not
illustrate any of the ring labs here. If interested, on the homepage of Al
Hibbard (http://www.central.edu/homepages/hibbarda/hibbard.html) you can find
a reference to the Exploring Abstract Algebra with Mathematica page, on which
you can find illustrations of ring labs.
What follows gives just a brief look at what can be done on the ring side.