Massey Exercise 3.4.4
Theorem: Let and be groups. If an element of has finite order, then it is either an element of or , or conjugate to an element of or .
Proof: Let . We induct on the length of .
If , then clearly, must be an element of or an element of , so the property holds trivially.
Suppose the property holds for all for some . Let be of length , so we have , and suppose has finite order – that is, for some . Observe the powers of :
It’s clear that if and are from different groups, has infinite order, because no simplification can occur in any power of . So it must be the case that and are in the same group. Then, let the product and note that:
Which is of length . So, by the inductive hypothesis, its conjugate to an element of or :
For some and . Putting it together, we have:
So is conjugate to an element of or (and this is true for any arbitrary word of length ).
It follows by induction that any word with finite order is conjugate to an element of or .