Pascal's Triangle and Its Relatives

Exercises

Pascal's triangles of some finite groups

1. Write the group table for Z2 x Z2. Verify that every element has order 1 or 2. Answer

2. Place (1,0) in all the boxes on the left side of the triangle template, and (0,1) in all the boxes on the right side. Fill in the rest of the triangle using the group table.
    Shade the boxes containing (0,0).
    Shade the boxes containing (1,0).
    Shade the boxes containing (0,1).
    Shade the boxes containing (1,1).
    Shade the boxes containing (0,0) or (1,0).
    Shade the boxes containing (0,1) or (1,1).
    Shade the boxes containing (0,0) or (0,1).
    Shade the boxes containing (1,0) or (1,1).
Repeat this exercise putting different elements along the left and right edges. What patterns do you see?
Answer

3. Now study the Pascal's triangles for the group D3 of symmetries of the equilateral triangle.
Here a denotes the identity symmetry, b and c are rotation about the center by 120 deg and 240 deg, and d, e, and f are reflection across the perpendicular bisectors of the base, right side, and left side. Write the table for this group.
Answer

4. Generate the Pascal's triangle placing b along the left side and d along the right side.
    Shade all boxes containing a, b, or c.
    Now shade all boxes containing d, e, or f.
    Shade all the boxes containing d, e, or f for the Pascal's triangle with b along the left side and e along the right, and with b along the left side and f along the right.
Explain the pattern you will get shading a, b, and c if the triangle is generated by b and c, if the triangle is generated by d and e.
In the Pascal's triangle for Z6 we found patterns found in the Pascal's triangle for Z2 and for Z3.
In D3 we find a Pascal's triangle found in Z2. Can you find a Pascal's triangle found intfound in Z3?
Answer

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