Abelisering

Abelisering is a term used in abstract algebra to describe the process of modifying a non-abelian group or ring to make it abelian. This is often achieved by introducing a new multiplication operation that is commutative. For example, given a group G with multiplication operation *, one might define a new operation ° such that a ° b = a * b * b⁻¹ * a⁻¹. This new operation ° often results in an abelian group, meaning that a ° b = b ° a for all elements a and b in the set.

The concept of abelisering is particularly relevant when studying the structure of non-abelian groups. By abelizing a group, mathematicians can gain insights into its underlying properties, such as its commutator subgroup. The commutator subgroup, which consists of all elements of the form [a, b] = a * b * a⁻¹ * b⁻¹, plays a crucial role in determining the non-abelian nature of a group. The quotient group of a group by its commutator subgroup is always abelian, and this quotient group is often referred to as the abelianization of the original group.

In the context of rings, abelizing can involve constructing a new ring where the multiplication is commutative. This can be achieved by considering the quotient ring of the original ring by its commutator ideal. The resulting quotient ring will have a commutative multiplication operation. Abelization is a fundamental tool for understanding the structure and properties of algebraic objects, particularly in relating non-abelian structures to their abelian counterparts.