Question

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

Answer 1

Answer:

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,

$ab \times bc = (ab)(bc) = a b^2 c = ac$, since $b^2 = 1$.  

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

Now, this is infinite, because we can have 'words' of $a, b, c, d, e, f$ of any length (the only simplifications we can use is when $a^2, b^2, c^2, d^2, e^2, f^2$ show up).

Therefore, any word (not containing $a^2, ..., f^2$) containing more than one of $a,b,c,d,e,f$ must have infinite order.