ibc said: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
The abelianization of a group is the process of transforming a non-abelian group into an abelian group by adding additional relations to the group's presentation.
The abelianization of a group G is calculated by taking the quotient of G by its commutator subgroup [G,G]. This results in a new group with the same elements as G, but with all commutators equal to the identity element.
Calculating the abelianization of a group can reveal important information about the structure and properties of the original group. For example, an abelian group is always solvable, so determining if a group is abelian can help in solving certain problems related to the group.
Not all groups can be abelianized, as some groups are inherently non-abelian and cannot be transformed into an abelian group by adding relations. However, many groups can be abelianized and it is a useful tool in group theory.
The abelianization of a group is closely related to its center, which is the set of elements that commute with all other elements in the group. The abelianization of a group is isomorphic to the quotient of the group by its center, meaning they have the same structure and properties.