```MODELING PERMUTATION GROUP

Mary Margaret Nee              Von Stueben Science Center
5039 N. Kimball Ave.
Chicago, IL 60625
1-312-588-2370

Objectives:
Throughout this unit the students are: to construct equilateral triangles on
cardboard; to develop the transformation concept with a one to one correspondence;
to generate the rotation and reflection table; and to deduce from the model the
abstract axioms of a group.

Behavioral Objectives:
The students will achieve practical knowledge of abstract concept of a group
and will verify this through testing with a desired goal of 85% success.

Materials:
1.  Large Plastic Model
a.  equilateral triangle with circumscribed circle
b.  square with circumscribed circle
2.  Transparencies
a.  Transformation
b.  Rotation and Reflection Table
c.  Algebraic axioms for a Group
B  Each Student needs
1.  Paper with a reference circle
2.  Cardboard Equilateral triangle
3.  Cardboard Square

Strategy:
A Background
There are many subject areas in Mathematics which would be enriched with this
presentation.  This should be introduced to a class where student cooperation
has been well established. The teacher must be extremely careful to instruct the
class in the correct way of formulating the transformations.

B Activity Outline
1.  pass out model triangles to students
2.  label the triangles in a clockwise way  A,B,C,
3.  model transformations on the overhead
4.  discover individually the finite elements under the definition
5.  create table of rotations and reflections
6.  observe patterns within the table
7.  name these patterns or laws
8.  define algebraicly the group
9.  verify the model with its laws as a group

C  General Plan
With the large equilateral triangle on the overhead  demonstrate the concepts
of rotation and reflection. Then allow the student to discover all finite
possibilities.
Once the elements are named start gnerating the table with the six elements.
From the observed data verify that closure, identity element, inverse for each
element and the associative axions or laws hold for the newly created table.
Continue to extend the binary operation to three elements by using the associative
2
law and then to any number of elements operating together.
Show on the overhead the list of definitions for an abstract group.  Again
affirm that the patterns created by the transformations of the equilateral
triangle form a set of elements which is well structured set of axioms with the
the binary operations of the transformation necessary for a group

1.  One to One transformation of Vertices
Name of Vertex          Transformed to Position
held by former Vertex

A  goes to  B
B    " "    C
C    " "    A         call this R1

A  goes to   C
B   "  "     A
C   "  "     B         call this R2

A  goes to  A
B   "  "    B
C   "  "    C          call this R3

A  goes to  A
B   "  "    C
C   "  "    B          call this ra

A   goes to C
B    "  "   B
C    "  "   A         call this rb

A  goes to  B
B    "  "   A
C    "  "   C         call this rc

2.  Table of Rotations and Reflections of a
Equilateral Triangle

"O"   R1   R2   R3   ra   rb   rc
_________________________________________
R1]     R2   R3   R1   rc   ra   rb
]
R2]     R3   R1   R2   rb   rc   ra
]
R3]     R1   R2   R3   ra   rb   rc
]
ra]     rc   rb   ra   R3   R1   R2
]
rb]     ra   rc   rb   R2   R3   R1
]
rc]     rb   ra   rc   R1   R2   R3
]

3.  Definitions
A.   Group "G" is a set of elements (a,b,c,d,e,...)
and a binary operation called  product "O" such
that the following axioms hold:
1.  Closure:    Form every ordered pair a,b of
elements of G the product a"O"b = c exists
3
and c is an elements of G.
2.  Associative Law: (a"O"b)"O"c  =  a"O"(b"O"c)
3.  Existence of Unity of the Identity element
An element "E" such that a"O"E = E"O"a =a
for every a an element of G
4.  Existence of Inverse of any element
For every a of G there exists any element
-1                    -1            -1
a   of G   such that  a  "O"a = a"O"a  =E

B.  Dihedral Group
The one to one mappings of a set onto itself
which preserves a property usually form a
group.

Symmetries of a geometric figure are of this
kind.  These are the congruent(distance-
preserving) mappings of the figure onto itself.

The symmetries of a regular polygon of sides
larger than or equal to  3  form a group
called  a Dihedral Groups of order 2n.

It can be shown within these dihedral groups
that
n
(Ri) =E  where E is the identity element
and i = 1,2,3,..(9n-1)
2          2         2
(ra) =E ,  (rb) =E,  (rc) =E or for any
reflection of the dihedral group squared

```