Return to Mathematics IndexMODELING PERMUTATION GROUPMary Margaret Nee Von Stueben Science Center 5039 N. Kimball Ave. Chicago, IL 60625 1-312-588-2370Objectives: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:A Overhead Projector 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 SquareStrategy: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 D. Overhead Transparencies 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