Answer:
[tex]<a, b, c, d, e, f | a^2 = b^2 = c^2 = d^2 = e^2 = f^2 = 1>[/tex]
Step-by-step explanation:
An infinite non-abelian group that has exactly six elements of finite order is,
[tex]<a, b, c, d, e, f | a^2 = b^2 = c^2 = d^2 = e^2 = f^2 = 1>[/tex]
Multiplication by concatenation can be done, for example,
[tex]ab \times bc = (ab)(bc) = a b^2 c = ac[/tex], since [tex]b^2 = 1[/tex].
This group is non-commutative, because [tex]ab[/tex] is not equal to [tex]ba[/tex] (as this is not a relation in the given presentation).
Now, this is infinite, because we can have 'words' of [tex]a, b, c, d, e, f[/tex] of any length (the only simplifications we can use is when [tex]a^2, b^2, c^2, d^2, e^2, f^2[/tex] show up).
Therefore, any word (not containing [tex]a^2, ..., f^2[/tex]) containing more than one of [tex]a,b,c,d,e,f[/tex] must have infinite order.
