- MacTutor History of Mathematics

Solution: Day 2, problem 2

Let the elements of finite order be numbered

t_{1}, ... , t_{N}

. We claim that any product

t_{i¬1} ... t_{i¬r} (1 ≤ i_{1}, ... , i_{r} ≤ N)

is equal to a product

t_{j¬1} ... t_{j¬r}

of the same length with

1 ≤ j_{1} ≤ ... ≤ j_{r} ≤ N

. This claim proves the result, since then every element of the group generated by

t_{1}, ... , t_{N}

belongs to the finite set

t_{1}^{\mathbb{Z}} ... t_{N}^{\mathbb{Z}}

and hence has finite order.
To prove it, we use induction on

r

, the case

r = 1

being trivial. Let

x = t_{i¬1} ... t_{i¬r}

. Since there are only finitely many representations of

x

t_{j¬1} ... t_{j¬r}

, there exists one with

j_{r}

maximal. By the induction hypothesis we may assume that

j_{1} ≤ ... ≤ j_{r-1}

. But the maximality of

j_{r}

and the representation

x = t_{j¬1} ... t_{j¬r-2} s t_{j¬r-1}

, where

s = t_{j¬r-1} t_{j¬r} t^{-1}_{j¬r-1}

is an element of finite order and hence equal to

t_{j}

for some

j

, together imply that

j_{r-1} ≤ j_{r}

, so we are done.