MA 1002 Worksheet 1 Due Date: September 16, 2009
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Finite Groups and Cayley Tables
Some groups are infinite. The group of integers under addition is an infinite group. There are an
infinite number of integers so the addition group of integers has an i
The Symmetry Group of the Triangle
There are six motions that can bring an equilateral triangle back into its original position. They
are
Do nothing
Rotate 120 degrees counterclockwise
Rotate 240 d
6. Subgroups
In the Atayun-HOOT! group we can limit our commands to two: "Attention!" and "About
Face!" That would give us a group with the Cayley table as follows:
*A B
AA B
BBA
But we have seen this
7. Cosets
Consider the subgroup of integers divisible by 3. This forms a subgroup of the additive group of
integers. Its elements are cfw_ . . . -9, -6, -3, 0, 3, 6, 9, . . .. By adding 1 to any multi
8. Lagrange's Theorem
Lagrange's theorem is the first great result of group theory. It states
THEOREM: The order of a subgroup H of group G divides the order of G.
First we need to define the order of
9. Cyclic Groups and Subgroups
Let's start with the number 1. We'll allow ourselves to add or subtract the number 1 to get to new
numbers.
Question: what integers will we be able to reach by this proc
10. Permutations
The "old shell game" is an example of a permutation. Three shells, one containing a pea, are in a
row in front of the sucker - er, I mean, client. The operator then changes the order
11. Permutation Groups
How many permutations are there on a group of n objects? Since there are n possible choices for
the first position and for each of these there are n-1 choices for the second pos
12. Rubik's Magic Cube
Rubik's Cube
In the early 1980's a puzzle invented by Ern Rubik of Hungary captivated the world's
imagination. Millions of the puzzles were sold, television programs appeared de
13. Rubik's Cube Groups
If we consider all sequences of moves on a Rubik's Cube we notice the following:
One sequence followed by another sequence is a sequence: Closure.
Followed by is associative.
14. Solving the Cube 1
From this.
.to this
The method that we will develop for solving the cube isn't the quickest or simplest one. In fact I
will streamline it a bit for practical use at the end of t
Group Housekeeping Theorems
The first step in the formal theory of groups is to take care of some details. In this lesson we'll
prove some basic properties of groups that we will use later on.
Theorem
Examples of Groups
There are an embarassing number of examples of groups. The most familiar ones come from
elementary arithmetic. The Integers form a group under the operation of addition. 0 is the
id
What is GROUP THEORY?
We'll throw some light on the title question of this page by asking another question. What is the
solution of the equation
(1)
4x = 3
The answer depends on what "things" we allow
MA 1002 Worksheet 2 Due Date: September 24, 2009
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MA 1002 Worksheet 3 Due Date: Seotember 30, 2009
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MA 1002 Worksheet 4 Due Date: October 7, 2009
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MA 1002 Worksheet 5 Due Date: October 7, 2009
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MA 1002 Worksheet 6 Due Date: October 21, 2009
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MA 1002 Worksheet 7 Due Date: October 28, 2009
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MA 1002 Worksheet 8 Due Date: November 4, 2009
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MA 1002 Worksheet 9 Due Date: November 11, 2009
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MA 1002 Worksheet 10 Due Date: November 18, 2009
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MA 1002 Worksheet 11 Due Date: December 2, 2009
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MA 1002 Worksheet 12 Due Date: December 9, 2009
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