Give an example of an infinite non abelian group that has exactly six elements of finite order

Step-by-step explanation:

An infinite non-abelian group that has exactly six elements of finite order is,  

Multiplication by concatenation can be done, for example,

, since .  

This group is non-commutative, because is not equal to (as this is not a relation in the given presentation).

Now, this is infinite, because we can have 'words' of of any length (the only simplifications we can use is when show up).

Therefore, any word (not containing ) containing more than one of must have infinite order.