ABSTRACT ALGEBRA
The resources in this part of the site could be described as manifest
resources for mathematicians. They have been used with undergraduate
mathematics majors at
For students having difficulty with concepts that seem remote, this more manifest approach has proved to be especially helpful. However it has proved even more useful for the students who can master remote concepts more easily. Many of these students who have gone on to traditional graduate programs in mathematics have said that the extra perspective obtained by exploring these resources gave them an advantage over students whose understanding was more remote.
Titles and descriptions of these resources are given below. Clicking on the title will allow it to be downloaded as a Microsoft word file. Many of these resources are only partially developed, and all of them could benefit from further development. I would especially like to include more examples and exercises. Most of these resources are individualized constructivist learning resource For a discussion of what is meant by an individualized constructivist learning resource and other aspects of concept construction, see the paper Constructivist Learning Resources.
Discussion about any of these resources and inquiries about others would be highly appreciated.
Basic Group and Ring Laws This paper examines some imaginary students deducing some basic propositional laws for groups and rings with the help of there mentor, a graduate student in Mathematic named Jo.
Finding Groups This is an individualized constructivist learning resource. It is designed to help a learner bridge the gap between group concepts that seem manifest and those that seem remote. It is intended primarily for a learner who finds the prerequisites below fairly manifest but feels that cosets and LaGrange’s Theorem are somewhat remote.
Magic Squares and Group Orbits This link takes you to a paper by John Pais and F Richard Singer which is published in the online journal Visual Mathematics. This paper is an essay in visual mathematics. It is a guided discovery in which the learner constructs his/her own math concepts, first intuitively, exploring, constructing, and counting 4 by 4 visual magic squares, and second analytically, using numerical representations and group orbits to classify and generate these visual magic squares.
Factorization Concepts.htm or Factorization Concepts.doc This resource develops the concepts of a principal ideal domain and a unique factorization domain, using a variety of specific examples.
Finding Groups This resource consists of problems in which you are to find all finite groups G that satisfy certain conditions. These conditions may determine exactly one group or several groups or even no group. This resource does not assume any of the main theorems about groups. The activities are designed to suggest conjectures about such results.
Finite Cyclic Rings This is a compact summary of my personal perspective on the main results involving modular structures and the relation of these results to the concepts of contemporary algebra. While I wrote it primarily as a resource for myself, I hope it might be useful to either teachers or students who want to supplement the study of abstract algebra with a study of specific structures. The exposition includes only main ideas and results. I have also included sample exercises and problems. The exercises and their answers include many of the details, including detailed proofs, that are only sketched in the main exposition. They also include some activities that helped me discover and organize my understanding of the algebra of modular structures. Parts of this resource are still being edited.
Finite Fields Results about finite fields and their extensions follow immediately from the general results on field extensions that can be in standard books on abstract algebra. This present resource is a more manifest approach to some a limited version of these ideas.
Galois Concepts & Unsolvability Results This resource is in an early stage of development, and I would like help in its further development. My minimal goal is to provide a resource that can be used to appreciate the unsolvability results that are suggested by material taught in secondary school mathematics. Furthermore I would like this resource to seem as manifest as possible to anyone who teaches secondary school mathematics. Thus instead of first developing theorems from Galois Theory (altho these are developed in later sections), this resource gradually introduced some galois concepts using a variety of examples. Each of these examples is augmented by a number of exercises that are intended to help the reader see in detail how these concepts relate to some specific fields. These examples and exercises should be sufficient to provide an intuitive perspective on the unsolvability results. Supplements provide proofs
Nim Group The main purpose of this resource is to develop a group that can be used to analyze a winning strategy for the 3 row nim game.
Pell Rings This resource begins by asking about the positive integer roots the equation: x2- 2y2 = 1. The strategy used to systematically generate these roots provides a sequence of fractions converging to Ö2. Altho the search is for integer roots, irrational factorizations are used. In the process results in a criteria for finding all the resources in the ring Z[Ö2]. This is generalized for other Pell Rings.
Permutation Groups This resource takes a highly manifest approach to determining all subgroups of the symmetric group on the set {0,1,2,3,4}. It presupposes a knowledge of all groups of order 11 or less, the theorem of LaGrange & Cauchy, the group of permutations for {0, 2, ..., n-1}, the use of cycle notation, the ability calculate product of permutations.
Permutation Puzzles This resource is a simplified version of the first part of the resource Permutation Groups. It provide simple examples and applications of an algebraic structure which we classify as a non-commutative group. Specifically it introduces the concept of a permutation and of a group of permutations. It then applies these concepts to the solution of various puzzles.
Polynomials Over Z6 Let f = x2+3x+2 as a Polynomials in Z6[x]. Since f = (x+1)(x+2) and f = (x+4)(x+5), this might suggest that f has two essential different prime factorization in this ring of polynomial. However it is possible to further factor {x+1, x+2, x+4, x+5} into pairs of factors of degree 1. We can, giving f = (2x+1)(3x+2)(4x+1)(3x+1). To see that this is a prime factorization, we need to show these factors are prime. We also need to show they have no higher degree factors. Altho the other factors of f in this table are associates of these factors, to see that this is essentially the only prime factorization of f, we also need to show that f has no higher degree factors. Note that f also factors as the product of two quadratics (2x2+1)(3x2+3x+2), but neither of these factors is prime.
Think-a-Dot Group The purpose of this resource is to develop a group structure D that can be used to analyze questions related to Think-a-Dot puzzles. It assumes you have solved some of these puzzles using Don Love's Think-a-Dot program The main mathematical background for this resource is the concepts of a group of bijective functions under composition, group generators, direct sum, morphism. This resource contains an appendix relating the analysis of D to the fundamental decomposition theorem for abelian groups. Generator conditions for D are given without reference to the Think-a-Dot device, and used to decomposed D as Z8Å Z2Å Z8. We then indicate how this idea can be generalized. A reader who is only interested this particular topic can be read the appendix without reference to the rest of the paper. Below is a very limited version of one section from of this resource. It merely illustrates a strategy which can be used to solve all of his puzzles, altho with a limitation discussed later. It also suggests further considerations about Think-a-Dot puzzles which are analyzed in the main resource. If you are interested primarily in a mathematical analysis you may want to skip this and just download the resource.
Square Roots in
Cyclic Fields A cyclic field is a field whose resources form a cyclic group
i.e. it is the filed of integers modulo p for some positive prime integer p.
Thus this resource involves taking using concepts from abstract algebra to
study what has been called quadratic residues in number theory. For example we
focus on the perfect squares in Zp as a subgroup of index 2 in the
group of resources for Zp. An intuitive perspective on the proof of
the quadratic reciprocity theory is portrayed visually using Z17 and
Z11.
Sylow Theorems This resource is designed to help a learner bridge the gap between group concepts that seem manifest and those that seem remote. It is intended primarily for learners who find many groups concepts fairly manifest but feel that the Sylow Theorems are somewhat remote. The main concepts that the learner needs to find manifest are those from the Finding Groups resource and those from the Permutation Groups resource.
THINK-A-DOT
GROUP STRATEGY
The Think-a-Dot Device: Recall that this device has 3 holes at the top thru which the red marble can be dropped. It then falls thru gates inside the device, each set to send the marble left or right. A marble does not hit a gate in level 2 when dropped thru a side hole whose gate at level 1 is set toward the wall. Unless otherwise specified, it should be understood that we have started with the initial pattern in which all gates set to the left. As a marble goes thru a gate it sets the gate in the opposite direction. On the outside in front of each gate is a blue dot if the gate is set left or a pink dot if it is set right. Let f, g, h denote the marble drops thru the left hole, middle hole, right hole respectively. The diagram below shows what you would see f followed by g followed by h.
|
n |
|
n |
|
n |
|
|
|
n n n |
f ® |
n n
n |
g ® |
n n
n |
h ® |
n n
n |
The unseen inside gate settings are as follows.
|
/ / / |
f |
\ / / |
g |
\ \ / |
h |
\ \ \ |
Since any setting can be in one of 2 states, it is convenient to think in terms of bits rather than colors. A zero indicates that a gate is set so a marble goes left. A one indicates that it is set so a marble goes right. Thus you can think of the result of applying these marble drops in terms of the bit patterns below.
|
0 0 0 |
f |
1 0 0 |
g |
1 1 0 |
h |
1 1 1 |
Notation: We use additive notation for marble drops. Thus the above sequence of drops is denoted as f+ g+ h. We also use multiples for repeated drops thru the same hole i.e. g+ 2f means drop thru g then two drops thru f. You might want to verify that the bit pattern which results from applying the sequence 2h+ f+ 2g would be top 100, middle 10, bottom 010.
Strategy: We now describe a simple strategy for making any possible pattern x. It is based on the observation that we can obtain any bit pattern in level 1 using some subset of {f, g, h}, that further use of an even number of drops in a hole will not alter any bit setting in level 1, nor will 4 drops in a hole alter any bit setting in level 2.
Gate Change Details: Starting with any pattern, 2f changes the left 1st level gate twice and the left 2nd level gate once. Thus 4f changes these gates an even number of times. 4f also changes the left 3rd level gate 3 times and the middle 3rd level gate 3 once. Starting with any pattern, indicate the number of times each gate is changed by 4g. Do the same for 4h.
We suggest you use these observations about gate changes to solve some puzzles and try to discover and describe a general strategy before the example below.
Example: Four drops thru the same hole changes gates only in level 3. This observation can be used to make the pattern with level 1 as 111, level 2 as 00, level 3 as 010. This is sketched and explained below. To obtain the desired first level of 111 we need to change each bit, so we begin with f+ g+ h. After this the second level of 11 differs from the desired second level of 00 in both bits. 2g leaves level 1 alone but reverses both bits in level 2. After f+g+h+2g the third level is 001, which differs from the desired bottom level 010 in its last 2 bits. 4h changes these 2 bits but leaves levels 1 and 2 alone. Thus d = f+g+h+2g+4h will give pattern x.
|
0 0 0 |
f+
g+ h |
1 1 1 |
2g |
1 1 1 |
4h |
1 1 1 |
Before reading our description of the general strategy you might try to apply a strategy like the above to some of the puzzles in Don Love's program. Then see if you can describe a general strategy.
Top Down Strategy: The strategy just used for 111 00 010 can be used
to find solve any of the puzzles given in Don Love's think-a-dot program. First
use single drops to obtain the desired level 1. If level 2 needs to be changed
select an element of {2f, 2g, 2h}. If needed, select an element of
{4f, 4g, 4h} to obtain the desired level 3. In more detail, let x be a pattern
you are to obtain.
(1) Create a pattern x1 by using nf+ kg+
mh where n, k, m are the left, middle, right entries in
the first level of x.
(2) Create a pattern x2 from x1
as follows: Let y be the bit by bit Z2 sum of the second levels
of x and x1. If y = 00 do nothing. If y = 10 use 2f. If y = 11 use
2g. If y = 01 use 2h.
(3) Create the pattern x from x2 as
follows: Let z be the bit by bit Z2 sum of the third levels
of x and x2. If z = 000 do nothing. If z = 110 use 4f. If z = 101
use 4g. If z = 011 use 4h.
Limitations: To find a sequence of drops by the top down strategy, we first choose at most 3 drops, then at most 2 drops, then at most 4 drops; giving at most 9 marble drops. The pattern 111,00,010 is an example in which using this strategy involves for 9 drops, and it can be shown that this pattern cannot be obtained with fewer than 9 marble drops. However that there are patterns for which the top down strategy does not yield the minimal number of marble drops. However this strategy is also limited because the marble drops needed can only be determined as you use them. The Think-a-Dot Group resource uses the mathematical concept of a group to find a strategy that will give the minimal number of drops without this limitation. This resource also gives a mathematical analysis that answer the question below and solves the following problems.
Question: Does the order in which drops are made affect to pattern?
Problem 1: Give a simple criteria for determining when a pattern is possible. For any possible pattern x, determine a sequence of marble drops that produces x. Find the minimum number of drops needed and a minimal sequence that produces x. Given any other natural number j determine if x can be obtained using a sequence of exactly j marble drops. If so, give such a sequence.
Problem 2: Let y and x be any patterns. Determine if there is a sequence of drops that changes y to x. If so find such a sequence. Also find the smallest number s of drops needed to obtain x from y a sequence of length s that does this. Given any other natural number j determine if x can be obtained from y using a sequence of exactly j marble drops. If so, give such a sequence.