MA 1002 Worksheet 1 Due Date: September 16, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer withou
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 infinite number of elements.
Some groups are finite. The
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 degrees counterclockwise
Flip about the symmetry axis t
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 little table before. It was just the northwest corner
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 multiple of 3 we get
another subset of group of integers, cf
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 a group or subgroup
Definition: If G is a finite group
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 process?
Answer: all of them.
To get to 17 simply add 1 16
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 of the shells
and challenges the client to tell which o
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 position and for each
of these n(n-1) ways of choosing the
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 devoted to it and
the medical world added "Rubik's thumb,
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.
The do nothing sequence does nothing: Identity.
Ever
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 the chapter. Our main emphasis however,
will be using gr
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 3.1: The identity element of a group is unique.
First
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
identity and the inverse of an element is called its nega
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 x to be. If we are doing all our arithmetic using
the
MA 1002 Worksheet 2 Due Date: September 24, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer withou
MA 1002 Worksheet 3 Due Date: Seotember 30, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer withou
MA 1002 Worksheet 4 Due Date: October 7, 2009
Print Name: Signature: ID #: Instructor/Section: Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without e
MA 1002 Worksheet 5 Due Date: October 7, 2009
Print Name: Signature: ID #: Instructor/Section: Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without e
MA 1002 Worksheet 6 Due Date: October 21, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
MA 1002 Worksheet 7 Due Date: October 28, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
MA 1002 Worksheet 8 Due Date: November 4, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
MA 1002 Worksheet 9 Due Date: November 11, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
MA 1002 Worksheet 10 Due Date: November 18, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer withou
MA 1002 Worksheet 11 Due Date: December 2, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
MA 1002 Worksheet 12 Due Date: December 9, 2009
Print Name: Signature: ID #: Instructor/Section:
Instructions: Explain each step using the conventions from class. Please mark solutions clearly. You will not receive full credit for a correct answer without
Bisnelly santana
11-3-15
ritical thinking page 358-359 motivating employees a monster of a problem
C
Economic factors affect businesses hiring approach because if the economy is
doing bad and unemployment is high they can get labor at a cheaper price
bec