301 Abstract Algebra
2:00 - 2:50 MWF
Hentzel 432 Carver
hentzel@iastate.edu
Web Page  http://www.math.iastate.edu/hentzel/class.301

Text: A First Course in Abstract Algebra, sixth Edition 
by John B. Fraleigh

Course: Chapter 1,2,3,4,5 

The material we are covering today and Friday is on 
Page 197 to 208 Section 2.3 beginning with 
"Group Action on a Set."  On Friday we will do
the theory which is essentially what is given in
the text.  Today we give the application.  This
is not covered very well in the text as far as
I can tell, but it is what he is trying to present in 
the five examples on pages 206-207.  

Main Idea:  Counting the various distinct objects.
Key Words:  Fixed.
Objectives:  Counting the number of distinctly colored cubes,
             necklaces, and place mats.

If one has a bracelet with six beads on it.
Then the following two bracelets are the same       

    X-------0                 0-------0                 
  /          \              /          \ 
 /            \            /            \
0              0          0              X 
 \            /            \            / 
  \          /              \          / 
    0-------0                 0-------0

But the following two bracelets are different.

    X-------X                 X-------0                 
  /          \              /          \ 
 /            \            /            \
0              0          0              X 
 \            /            \            / 
  \          /              \          / 
    0-------0                 0-------0
The reason is that one can twist the first into the
second in the top case, but no amount of twisting
will deform the bracelet so that positions which
are together will move apart.  

      In a situation like this, we are given a group
which are the actions allowed.  If two objects can
be adjusted by the group till they are identical, then
we call those objects identical.  The problem is to
count how many different objects there are.

This is a numerical lecture.  The proofs that what you
are doing really works will come on Friday.

How many distinct bracelets with 6 beads can be constructed
using 2 colors of beads.

    1-------2
  /          \ 
 /            \
6              3 
 \            / 
  \          / 
    5-------4

The group   I  (123456) (135)(246) (14)(25)(36) (153)(264) (165432)
corner flips   3 of them of type   ( )( )( , )( , )
edge flips     3 of them of type   ( , )( , )( , )  
                                                  left fixed
The group has    I                              1  64      = 64
              2  ( , , , , , )                  2   2         4
              2  ( , , )( , , )                 2   4         8
              3  ( )( )( , )( , )               3  16        48
              4  ( , )( , )( , )                4   8        32
            -----------------------                      --------
             12  elements total.                            156
 
Number of necklaces is 156/12 = 13.

Once the bracelet is on the wrist, how many are there?

Group is:   I (123456) (135)(246) (14)(25)(36) (153)(264) (165432)
Fixed:     64    2         4            8          4          2

         84/6 = 14

This means that one bracelet has two distinct ways to wear it, depending
on which way it is slipped over the arm.  Which is it?  

                ##################### 
                #...................#
                #.........|.........#
                #...red...|...blue..#
                #.........|.........#
                #_________|_________#
                #.........|.........#
                #..green..|...blue..#
                #.........|.........#
                #.........|.........#
                ##################### 
                   
How many place mats are there using 3 colors?

Symmetries of the Square
     I ( )( )( )( )                   81
     2 ( , , , )                       6
     2 ( , )( )( )                    54
     3 ( , )( , )                     27
   -------------                    -----
     8                               168

168/8 = 21

What are they.

      Same color  3-1         2-2         1-1-2
        AA         AA       AA   AB    A B    A C
        AA         AB       BB   BA    C C    C B
         3          6        3    3     3       3


What is the group of symmetries of a Cube?
How many elements are there and what are their
cycle structure.

Problems:

(1)  How many ways can you paint the faces of a cube with 2 colors?

(2)  How many ways can you paint the faces of a cube with 3 colors?

(3)  How many 8 bead necklaces are there with 2 colors?

(4)  How many 8 bead necklaces are there with 3 colors?