# Primitive Elements in the Free Group of Rank Two

Primitive Elements in the Free Group of Rank Two

This Demonstration determines whether or not a word in the free group of rank two is primitive. Let and . The word is first cyclically reduced, that is, and so on. If it contains both and or and , then the word is not primitive. Otherwise, the word is written along the unit circle moving between letters, where is the length of the word and , where and are the absolute values of the exponent sums of and , respectively. A red disk indicates and a blue disk indicates . If either of the two letters occupy the same location on the circle, or if the letters are not in two contiguous groups, the element is not primitive.

A=a

-1

B=b

-1

axA=x

a

a

-1

b

b

-1

2πk

w

w

k=min{w,w}

a

b

w

a

w

b

a

b

a

±1

b

±1