Calculate the abeliazation of the group G=<a,b>

[ibc]

Does anyone know how can I calculate the abeliazation of the group G=<a,b>
(when a and b are totally free and independent)
i.e. first of all, how do I calculate the commutator group [G,G]?
and then how do I find the quotient group: G/[G,G]

Thanks

 

[wofsy]

Does anyone know how can I calculate the abeliazation of the group G=<a,b>
(when a and b are totally free and independent)
i.e. first of all, how do I calculate the commutator group [G,G]?
and then how do I find the quotient group: G/[G,G]

Thanks

I would just count the number of generators in the abelianization then appeal to two facts: the abelianization has no elements of finite order and a free abelian group is a sum of copies of the integers.

Or you could directly show that a and b commute in the quotient group and that there are no other relations - e.g. each generate a free abelian group on one generator. For instance the powers of a and the powers of b are not commutators.