Pascal's Triangle and Its Relatives

Exercises

Symmetries of regular polygons

4. First say the 2n words of the form

1, a, a2, ..., an-1, b, ba, ba2, ..., ban-1

are canonical words.

Now we show that any word can be reduced to one of these canonical words by applications of the relations an = 1, b2 = 1, and (ab)2 = 1.

Using b2 = 1 we can replace every every occurrence of bk in w by b (if k is odd) or 1 (if k is even).

Next, abab = 1 is equivalent to bab = a-1, in turn equivalent to ba = a-1b. With this rule, all occurrences of b can be moved to the left.

With this, w is equivalent to biaj. Finally, i can be reduced to 0 or 1, and j to 0, 1, ..., n-1.

Return to Symmetries of Regular Polygons Exercises.